Papers

A key theme in modern extremal combinatorics is the study of classical combinatorial theorems relative to sparse subsets of their natural settings. Here we describe some of the recent progress in this area and state a number of problems that remain open and pressing.

We give a quantitative combinatorial characterization of size-oblivious one-sided testability in the dense graph model, resolving a question of Alon, Fischer, Newman, and Shapira. For hereditary graph properties, we prove that one-sided testability is quantitatively equivalent to the existence of suitable hypergraph containers, a central and widely used tool in modern combinatorics. Combining this equivalence with the Alon-Shapira notion of semi-hereditariness yields a quantitative characterization of arbitrary graph properties. The correspondence is effective in both directions and provides explicit translations between tester complexity and container parameters. Our proof is regularity-free and extends uniformly to every fixed finite relational signature of bounded arity, including digraphs, coloured graphs, and hypergraphs. As applications, we obtain quantitative closure results for partition properties and testers for properties defined by the existence of a linearly large induced substructure.

In this paper, we generalize the Steenbrink vanishing theorem for ample line bundles on complex projective varieties by extending it to big line bundles on compact complex spaces with multiplier ideal sheaves.

We study the computability of Julia sets for polynomial diffeomorphisms of $\mathbb{C}^2$ with dynamical degree $d>1$, whose prototypical examples are complex Hénon maps. In previous work, we established computability under the assumption of hyperbolicity (Axiom A). Here, we extend this result to maps whose Fatou components are attracting basins, allowing for the possibility of no attracting basins or infinitely many basins. This yields computability of the Julia set for several classes of non-hyperbolic maps, including certain substantially dissipative maps in the Lyubich-Peters class, and certain quasi-hyperbolic maps. Our proof is based on an algorithm that separates the dynamics into escaping, attracting, and saddle regimes. A key ingredient is the use of stable manifolds of saddle periodic points to approximate the forward Julia set via backward iteration. We first prove the result for generalized Hénon mappings and then extend it to arbitrary polynomial diffeomorphisms by expressing them as finite compositions of generalized Hénon maps. Finally, we present examples of non-computability in the presence of neutral dynamics, including Hénon maps with computable coefficients exhibiting semi-Siegel behavior. These examples show that the computability/non-computability dichotomy associated with neutral dynamics in one-dimensional complex dynamics in part persists in higher dimensions.

Let $\mu$ be a log-concave probability measure on $\mathbb R^n$ and let $f\colon\mathbb R^n\to\mathbb R^k$ be a polynomial mapping of degree at most $d$. We show that \[ \mu(f\in A) \le C\bigl(\lambda_k(A)\bigr)^{\frac{1}{k(d-1)+1}} \] for every Borel set $A\subset\mathbb R^k$ whenever the image measure $\mu\circ f^{-1}$ is absolutely continuous. The constant $C$ is independent of the dimension $n$, and the exponent $\frac{1}{k(d-1)+1}$ is sharp. This extends the scalar Carbery–Wright inequality and answers, in the log-concave setting, a question raised by Avni, Glazer, and Larsen. In addition, we show that the density of $\mu\circ f^{-1}$, whenever it exists, belongs to the Nikolskii–Besov space $B^{\frac{1}{k(d-1)+1}}_{1,\infty}(\mathbb R^k)$, with a dimension-free bound for the corresponding norm. A central difficulty in passing from scalar polynomials to vector-valued polynomial mappings is the lack of a suitable nondegeneracy parameter quantifying absolute continuity of $\mu\circ f^{-1}$, as the variance does in the scalar case. Natural candidates such as the covariance matrix or the Jacobian matrix either fail to characterize this property or do not lead to dimension-free estimates. We identify such a parameter and define it to be the covariance matrix of the vector formed by the monomials of degree up to $d^{k-1}$ in the normalized components of $f$. The dimension-free nature of our results allows us to extend Kusuoka's absolute continuity criterion for Gaussian polynomial random vectors to the log-concave setting. Moreover, in this setting, we obtain estimates relating convergence in distribution to convergence in total variation for polynomial random vectors.

Aumann (1974) showed that an atomless public randomization device makes the feasible- and equilibrium-payoff sets of a game with a fixed information structure convex, and asked whether they are closed. We show that, in every case the question leaves open, they need not be. One information structure drives all the examples: two sequences of fair signs whose coordinate correlations increase to a ceiling $\rho<1$ that no pair of separately measurable square-integrable rules attains. For every $0<\rho<1$ it yields a three-player game with a public randomization device whose equilibrium-payoff set is exactly the open interval $\{(0,0,t):-\rho<t<\rho\}$; a two-player game with a public randomization device whose equilibrium-payoff set is convex, full dimensional, and not closed; and, without any public device, nonclosed feasible- and equilibrium-payoff sets, the latter along equilibria with unique best replies modulo null events whose payoffs approach a vector that is not even feasible. With distinct but mutually absolutely continuous subjective priors, even the feasible-payoff set can fail to be closed in the presence of an objective public randomization device, together with every $\varepsilon$-equilibrium payoff set and the set of induced law tuples. Our construction also allows us to resolve a conjecture of Stinchcombe (2011). The main results and the lemmas supporting them are formalized in the Lean proof assistant; an appendix records the exact coverage of each statement, including the clauses for which only a paper proof is given.

We prove a quasiflat closing theorem and a coarse flat torus theorem for hierarchically hyperbolic groups (HHGs). Namely, given an HHG $G$, we prove that $G$ is hyperbolic if and only if it contains no $\mathbb Z^2$ subgroups and, if $A\leq G$ is virtually $\mathbb Z^n$, then there is an $A$–invariant $n$–dimensional uniform quality quasiflat $F$ such that any two points in $F$ are joined by a uniform-quality hierarchy path lying in $F$. The later is a consequence of a more detailed theorem describing a “coarse minset” for $A$ in $G$, which has various applications, including an ascending chain condition for virtually abelian subgroups, hierarchical quasiconvexity of highest abelian subgroups, and some geometric control over normalisers, centralisers, and commensurators of abelian subgroups. We use this to rule out HHG structures for certain Coxeter groups on the basis of their affine subgroups, and to give a new proof that virtually solvable subgroups of HHGs are virtually abelian, which simplifies the original proof by avoiding Gromov's polynomial growth theorem.

We prove that a general Brill–Noether curve $C$ of genus $g \geq 2$ and degree $d$ in $\mathbb{P}^4$ has stable normal bundle $N_C$ if and only if $$(g, d) \notin \{(2,6), (3,7), (5,8), (6,9), (7,10)\}.$$ Moreover, $N_C$ is strictly semistable if $(g, d) \in \{(3, 7), (5, 8)\}$, and is unstable if $(g, d) \in \{(2, 6), (6, 9), (7, 10)\}$. Our results are valid in any characteristic. Along the way, we also generalize previous results of Larson–Vogt on interpolation for $N_C(-1)$, from characteristic zero to arbitrary characteristic.

This paper studies the fully nonlinear Alt-Phillips free boundary problem \begin{equation} F(D^2u) = u^\gamma \chi_{\{u>0\}}, \end{equation} for $\gamma \in (-1/3,1)$ without any convexity assumption on $F$. We establish that flat free boundaries are smooth. This result is new even for the fully nonlinear obstacle problem when $\gamma = 0$. Our approach is based on a partial hodograph transform, which converts the free boundary into a fixed boundary and produces a fully nonlinear degenerate equation. For this equation we prove a Harnack inequality, yielding an improvement of flatness, and a Schauder estimate for the associated linearized equation. Both estimates appear to be new and are of independent interest.

For every $g\geq 3$, every closed, connected, oriented, simply connected smooth $4$-manifold $X$, and every knot $K\subset S^3$, we construct infinitely many pairwise topologically inequivalent smoothly embedded oriented genus-$g$ surfaces $F\subset X$ whose orientation-preserving extendable mapping class subgroups are trivial in both the topological and smooth categories and whose first Alexander modules are isomorphic to the Alexander module of $K$. In particular, there are infinitely many such surfaces with vanishing first Alexander module. The construction is supported in a $4$-ball. Although their Alexander data are prescribed independently, the surfaces are distinguished, and their mapping-class rigidity is detected, by the nonabelian centralizer structure of their exterior groups.

Let $G$ be a (multiplicatively written) finite group. A sequence over $G$ is a finite collection of terms from $G$, where repetition is allowed and the order is disregarded. A product-one sequence is a sequence whose terms can be ordered such that their product in $G$ equals the identity element of $G$. The set $\mathcal B (G)$ of all product-one sequences over $G$, endowed with the concatenation of sequences as the operation, is a finitely generated C-monoid; in particular, it is atomic, i.e., every non-unit element can be written as a finite product of atoms. The study of $\mathcal B (G)$ is of fundamental importance, as its combinatorial, algebraic, and arithmetic properties play a crucial role across various branches of mathematics, most notably in invariant theory and factorization theory. While the arithmetic of the monoid $\mathcal B (G)$ is well understood in the abelian setting (in which case $\mathcal B (G)$ is a Krull monoid), little is known in the non-abelian setting because of the substantial structural complexity involved. In this paper, we study the arithmetic invariants of the monoid $\mathcal B (G)$ for non-abelian groups, focusing in particular on the catenary degree. The catenary degree $\mathsf c (G)$ of the monoid $\mathcal B (G)$ is defined as the smallest integer $N$ such that any two factorizations of an element $S \in \mathcal B (G)$ can be concatenated by a chain of factorizations in which adjacent steps differ by replacing at most $N$ atoms. Extending the methods from arithmetic combinatorics to the non-abelian setting, we explicitly characterize all finite groups with catenary degree at most 3, and we investigate an infinite class of finite groups whose monoids of product-one sequences are seminormal and possess well-behaved arithmetic structures. Furthermore, we show that a specific non-abelian group in this class has catenary degree 4.

In this article we provide new bounds for the Mertens sums and products including $\sum_{p \le x} p^{-1}$ and $\prod_{p \le x} (1-\frac{1}{p})$ which provide superior exponential and log type bounds for these sums in all ranges. These weighted prime number sums and products were extensively studied by Rosser and Schoenfeld (1962) and are employed in a wide range of applications in number theory, cryptography, and combinatorics. Extensive tables are provided in this article which will be useful for these types of applications. The main new ideas in this article are sharp bounds for weighted sums of zeros of zeros of the zeta function. We make use of a novel technique of Fiori-Kadiri-Swidinsky (2023) which relies on a recent explicit zero-density estimate for $N(\sigma,T)$ of Kadiri-Lumley-Ng (2018). The bounds and techniques in this article for weighted zeros sums will likely be useful in many other arithmetic applications. Our main theorem significantly improves the exponential decay result of Vanlalngaia (2017) and fills a gap in the literature by correcting work of Dusart (2018). The results are also presented in a way that are amenable to future improvements. In addition, we prove an exact “Riemann-Guinand explicit formula" for the Mertens sum $\sum_{p \le x} p^{-1}$ that appears to be new.

We introduce a special mapping from pairs of power series to the group of Sprugnoli matrices. This mapping has the property when the second argument is an even power series, then the square of the resulting Sprugnoli array is an aerated element of the Appell subgroup of the Riordan group. This allows us to explore the square roots of elements in the aerated Appell subgroup. As the identity is an element of this subgroup, we are led to explore related involutions in the Sprugnoli group.

Let $K/k$ be a finite abelian extension of number fields with Galois group $G$ and let $n\geq 2$. We prove, assuming the relevant $p$-part of the equivariant Tamagawa number conjecture, first-derivative analogues of the Deligne-Ribet integrality theorem and of the Coates-Sinnott conjecture. We construct a rank-one leading term from the first derivatives at $s=1-n$ of the $S$-truncated Artin $L$-functions and show that it satisfies an integral annihilation property. We then attach to this leading term a fractional ideal of $\mathbb Q_p[G]$ and prove that, up to the natural torsion factor coming from $K_{2n-1}(O_K)$, this ideal annihilates the even $K$-group $K_{2n-2}(O_{K,S})$. The proof uses determinant methods, $\Sigma$-modified étale complexes, and a cancellation argument which removes the auxiliary Euler factors.

We consider degenerate principal series representations $\operatorname{Ind}_P^G\chi$ over finite fields, where $G$ is a classical subgroup of $\operatorname{GL}_{2n}$, and $P$ is the Siegel parabolic subgroup. For example, we show that this representation is always multiplicity-free and irreducible for generic characters $\chi$. We then discuss a particular intertwining operator $I$ on $\operatorname{Ind}_P^G\chi$ and its related combinatorics. Firstly, this operator $I$ produces families of diagonalizable antitriangular matrices with well-behaved eigenvalues. Secondly, applying $I$ to a special vector in $\operatorname{Ind}_P^G\chi$ leads us to various matrix Gauss sums, whose evaluations imply an explicit equidistribution result of the trace and determinant of symmetric and alternating invertible matrices.

We propose the VIX-derived volatility (VDV) model, a VIX-first framework for joint SPXVIX modeling. In the model, we define explicit dynamics for the VIX process to price VIX futures and options, yielding a VIX-side calibration that is independent of the SPX dynamics. Using the rolling-window definition of the VIX, we then derive a coupling function to obtain the SPX volatility process as a latent process consistent with the calibrated VIX dynamics. This produces a stochastic-volatility representation for SPX that can be further calibrated to SPX options without altering the VIX dynamics. Relative to existing joint-calibration approaches based on global optimization or highly flexible black-box dynamics, the method offers an interpretable and tractable decomposition of the joint problem by separating the calibration of the VIX from the calibration of the SPX. In a numerical experiment, we show that a VDV model with local volatility and mean-reverting dynamics for the VIX achieves a close model fit to the VIX futures market, VIX option market, and the SPX option market at the same time, providing a consistent joint framework.

{We study the module structure of the polynomial ring, localized at the discriminant, over the ring of differential operators on the ring of invariants of the imprimitive complex reflection group $G(r,p,n)$, describing its simple components with explicit generators given by the higher Specht polynomials of Ariki–Terasoma–Yamada. The proof rests on a Jacobian lemma computing the discriminant of $G(r,p,n)$, combined with a double-centralizer argument. As particular cases ($r=2$, $p=2$ or $p=1$) we recover, and considerably shorten, the known decomposition theorems for the real reflection groups $W(D_n)$ and $W(B_n)$; we also treat $G(r,r,n)$ and $G(r,1,n)$ explicitly, with worked examples ($D_2$, $D_3$, $B_2$) and their central idempotents. Finally, applying the Galois descent equivalence of categories of Nonkané to $G(r,p,n)$ for the first time gives a second, generator-free description of the simple summands as twisted invariants.}

We consider the boundary value problem $y''=\kappa^2y^n, y(0)=1, y(1)=0,$ relevant to radiative heat transfer, chemical reactions, etc. We use upper/lower envelopes to study the qualitative and quantitative dependence of its solution on parameters, and transform it into an easily solvable initial value problem.

Let $G$ be a higher-rank simple Lie group acting on a space $X$. A theorem of Nevo and Zimmer asserts that every ergodic stationary probability measure on $X$ is either $G$-invariant or admits a projective factor $G/Q$ for a proper parabolic subgroup $Q$. We develop a quantitative theory of stationary measures and prove an effective form of this dichotomy. We introduce notions of $\eps$-almost stationarity, $\delta$-almost invariance and $\delta$-almost projective factor, and show that every $\eps$-almost stationary measure is either $\delta$-almost invariant or carries a $\delta'$-almost projective factor, with $\delta,\delta'$ explicit in $\eps$ and depending only on $G$. No ergodicity, arithmeticity or Diophantine hypothesis is imposed, and the bounds are uniform over all $G$-spaces. Additionally, we prove a decomposition version of this theorem where each measure is decomposed into an almost invariant part and an almost projective part. This comes at the cost of worse constants. The proof introduces several tools: the entropigeonhole method, an entropy-based pigeonhole principle yielding a quantitative Mautner phenomenon; factor functions, quantitative analogues of functions on homogeneous factor spaces; and a fast generation dichotomy in the spirit of growth in groups. In a companion paper these are used to show, among other things, that a discrete subgroup of infinite covolume has injectivity radius at least $c\log^{(4)}r$ somewhere in the ball of radius $r$, which is an effective form of a theorem of Frączyk and Gelander. We also prove rates for Benjamini–Schramm convergence of quotients of higher-rank lattices.

We investigate the Emden-Fowler equation $u''=-x^ru^p, u(0)=1, u'(0)=0$, with roots in astrophysics, and study the qualitative and quantitative dependence of its solution on the parameters; the analytical work is paralleled by numerical simulations. A special attention is given to estimating the first zero $x_0$ of $u$ in terms of $r, p$. The results are used to address two apparently new issues: first, we solve EF backwards starting from $x_0$; second, we transform it into an overdetermined boundary value problem and decide when is this solvable.

A family of automorphic de Branges - Rovnyak spaces (depending on an arbitrary character $\alpha$) is associated to a $\beta$-automorphic analytic function $w$ bounded in modulus by $1$. Multiplication by the Green function defines unitary operators acting between those spaces. Function $w$ can be recovered from these operators as the characteristic function. It is shown that for Nevanlinna-Pick problem this family of unitary operators extends the family of isometric operators defined by the interpolation data.

We investigate the sharp regularity up to the boundary for viscosity solutions of fully nonlinear parabolic equations with oblique derivative boundary conditions given by \begin{equation*} \left\{ \begin{array}{rclcl} F(D^2u,x,t) - u_{t} &=& f(x,t) & \text{in} & Q_{1}^{+}, \\ \beta(x,t) \cdot Du &=& g(x,t) & \text{on} & Q_{1}^{*}. \end{array} \right. \end{equation*} The regularity theory is developed according to the integrability of the source term, the smoothness of the boundary data, and the oscillation of the coefficients of the operator \(F\), using a compactness method combined with polynomial approximation. In the borderline case \(f\in L^{n+2}\), we obtain Log-Lipschitz continuity of solutions up to the boundary. Under stronger integrability, specifically when \(f\in L^{p}\) for some \(p>n+2\), we establish optimal \(C^{1+\alpha',\,\frac{1+\alpha'}{2}}\) boundary estimates. At a higher regularity level, we prove Schauder-type estimates under appropriate assumptions on the operator \(F\) and the boundary data. As a byproduct, we obtain parabolic \(C^{1,\mathrm{Log\text{-}Lip}}\) regularity in the critical borderline case where the source term belongs to BMO space.

Many mixing dynamical systems $(X,T,\mu)$ are known to satisfy the hitting time statistics result \[ \lim_{r \to 0} \mu \{\, x : \tau_{B(y,r)} (x) > t/\mu(B(y,r))\,\} = e^{-t}, \] for $\mu$-almost every $y$, where $\tau_{B(y,r)} (x)$ is the first hitting time of $x$ to the ball $B(y,r)$. Taking a different point of view, we fix $x$ and consider $\tau_{B(y,r)} (x)$ as a function of $y$. We call this the visiting time of $y$ from $x$, i.e. the time it takes for $y$ to get a visit from $x$ within a neighbourhood of radius $r$. We prove that \[ \lim_{r \to 0} \mu \{\, y : \tau_{B(y,r)} (x) > t/\mu(B(y,r)) \,\} = e^{-t}, \] for $\mu$-almost every $x$. As a byproduct we obtain a new method of proof for hitting time statistics.

We study a neutral periodic orbit of a continuous two-field flow that is tangent to a switching seam while nearby orbits enter the second field. We separate the roles of two governing integers: the neutral-return order $n\geq2$ controls orbit count, whereas the even contact order $\nu$ controls grazing codimension, passage scale, and transverse crossover. Continuity factors the branch mismatch as $X^+-X^-=hW$; an active excursion of duration $O(q^{1/\nu})$ therefore produces a correction of order $q_+^{1+1/\nu}$. A parameter-uniform passage theorem and a cross-seam Hermite zero theorem then give the sharp fixed-stratum cyclicity: $n$ or $n+1$ periodic orbits, according to the sign coupling of smooth and active terms. A rank-$n$ physical unfolding attains the applicable bound. Contact-jet incidence gives ambient codimension $\nu-2$ for order-$\nu$ grazing and $n+\nu-2$ for simultaneous neutral grazing. Transverse contact parameters replace the central power by a weighted positive-part cap whose onset exponent crosses from $1+1/\nu$ to the generic quadratic-contact value $3/2$. The same sharp bound holds for both the exact cap and a prepared physical multiwell return, including births, mergers, and multiple entry–exit pairs. A value-only $C^1$ counterexample shows that finite cyclicity requires event-derivative information. For every even $\nu\geq4$, a closed polynomial family realizes the prescribed contact and rank conditions and the sharp periodic-orbit configurations through actual finite-time first-return maps.

We study coarse topology of nodal sets of linear combinations of eigenfunctions of sub-Laplacians. More precisely, we prove coarse versions of Courant's and Bézout's theorems for linear combinations of eigenfunctions of sub-Laplacians on compact nilmanifolds obtained as quotients of stratified groups. We conjecture the extensions of these results to general closed equiregular sub-Riemannian manifolds and outline a programme for proving them. The method we use combines topological persistence and anisotropic Sobolev theory of Hörmander vector fields, generalizing the ideas which have recently been implemented in the Riemannian case.

A seamless parametrization of a closed oriented surface carries a discrete invariant, its holonomy signature: the cone angles, all multiples of $\pi/2$, together with the rotational holonomy $\rho\colon H_1(M\setminus C)\to\mathbb{Z}_4$ of the induced cross field. This is the datum a quadrangulation prescribes, and it decides whether any parametrization exists at all. Shen, Zhu, Capouellez, Panozzo, Campen and Zorin asked which signatures occur and gave a sufficient condition of gcd type; which signatures are realizable has remained open. We answer the question. A Reduction Lemma shows that the mapping class group acts on signatures with fixed cone angles with orbits classified by the subgroup $\mathrm{im}\,\rho\le\mathbb{Z}_4$ alone, so at most three cases survive per angle multiset instead of $4^{2g}$. A dictionary then identifies seamless parametrizations with meromorphic 4-differentials, under which $\mathrm{im}\,\rho$ measures primitivity, and realizability becomes non-emptiness of a stratum of primitive $k$-differentials with $k=4/d$ and $\mathrm{im}\,\rho=\langle d\rangle$. Unwinding this against the known classification of such strata leaves exactly five exceptional families; every other admissible signature is realizable, in every genus. Two of the five appear to be new, and both live in genus two. Four of the five lie outside the gcd condition, and the whole region it leaves open is settled here. The non-emptiness half is made constructive by an explicit one-vertex square-tiled surface in every genus together with a local surgery that splits one cone into two of prescribed angles, leaving the genus, the other cones and $\mathrm{im}\,\rho$ untouched. Two extensions follow: surfaces with boundary, the feature-aligned setting, and the relation to the Abel-Jacobi criterion at a fixed conformal structure.

We prove a nine-distance theorem for Kronecker sequences on flat three-tori. That is, we show that among the first $N$ orbit points, at most nine distinct positive nearest-neighbour distances occur. This proves the conjecture of Haynes and Marklof. An example of Dettmann shows that nine is optimal. More generally, we prove that on a flat $d$-dimensional torus the number of such distances is at most $2^d+1$. The main tool is a new growth theorem for the denominators $q_1<q_2<\cdots$ of best simultaneous approximations in a $d$-dimensional inner-product space, which is of independent interest. We prove that, whenever $q_{n+2^d}$ is defined, either $q_{n+2^d}\ge2q_{n+1}$, or the indices $1,\ldots,2^d$ can be partitioned into disjoint pairs $\{j,k\}$, $j<k$, such that $q_{n+k}=q_n+q_{n+j}$. In particular, $$ q_{n+2^d}\ge \min\{2q_{n+1},q_n+q_{n+2^{d-1}}\}\ge q_n+q_{n+1}. $$

The X-ray of a permutation is a sequence of sums along each diagonal of the associated permutation matrix. They satisfy certain necessary constraints on distribution of the values, which are conjectured to be sufficient when the sequence is binary. By re-expressing the constraints in a form that allows for real-valued relaxations, we prove that these binary sequences are always X-rays of doubly stochastic matrices.

We generalize work of Popa-Schnell on Viehweg hyperbolicity for smooth projective varieties to the case of smooth Deligne-Mumford stacks. The results of Popa-Schnell build on results of Viehweg-Zuo, utilize a result of Campana-Păun, and extend results of Kebekus-Kovács and Patakfalvi. We show that if a smooth proper DM stack with projective coarse moduli space parameterizes a family of varieties of general type having maximal variation, then the natural log canonical bundle of the stack is big. We also consider implications for the coarse moduli space of the stack, and apply the results to several standard moduli spaces.

We prove local smoothing estimates for wave equations with $C^{r}$ coefficients, relying on bilinear restriction estimates associated with rough wave propagation. In two and three dimensions, for $C^{1,1}$ coefficients we recover the sharp $L^p$-$L^q$ local smoothing bounds previously established for smooth coefficients. These are implied by stronger $L^{p}$-$L^{q}$ decoupling inequalities with endpoint derivative loss. As a corollary, we obtain non-trivial $L^{p}$-$L^{p}$ local smoothing estimates as well.