Papers

Let $\mathcal{M}=h\mathcal{K}_u$ be a nearly $S^*$-invariant subspace of $H^2$, where $\mathcal{K}_u=H^2\ominus uH^2$ and $h$ is the extremal multiplier. For $\vp\in L^\infty(\T)$, we study the compression \[ D_\vp^\mathcal{M} = \restr{P_{\mathcal{M}^\perp}M_\vp}{\mathcal{M}^\perp}, \] called a weighted dual truncated Toeplitz operator. When $h\equiv1$, this reduces to the classical dual truncated Toeplitz operator. Using the Hartmann–Ross projection formula, we prove \[ \|D_\vp^\mathcal{M}\|=\|\vp\|_\infty, \] characterize compactness, and show that the natural multiplication map by $h$ identifies the weighted and classical theories precisely when $h$ is inner. We also establish complex symmetry and obtain block matrix, defect, and semi-commutator identities via weighted truncated Hankel operators. As a main algebraic consequence, we prove \[ D_\vp^\mathcal{M} D_\psi^\mathcal{M}=0 \quad\Longleftrightarrow\quad \vp=0\ \text{or}\ \psi=0 \quad\text{a.e. on }\T. \] Finally, we derive a a rank-at-most-two correction formula and a finite-rank displacement identity for the generalized dual shift $D_z^\mathcal{M}$.

We prove stable self-similar blowup for the harmonic map heat flow in dimensions 3 to 6 for target manifolds which are slightly perturbed versions of the round sphere. Starting from the known blowup solution for the sphere, we construct self-similar blowup solutions for this class of target manifolds and prove that these solutions are asymptotically nonlinear stable. Although the blowup solution for the sphere is not explicitly known in this case, we can still use perturbative methods similar to those developed by Donninger, Schörkhuber, and the author in \cite{DonSchWit26} who consider a related problem for the wave maps equation.

Aleman, Olsen, and Saksman asked whether radial convergence may fail for bounded analytic functions on the infinite-dimensional polydisc when every radial point has non-increasing coordinates, and whether the approach may be chosen independently of the boundary point. We construct a point-dependent counterexample in which every fixed coordinate nevertheless increases to $1$. A finite discretization yields a boundary-point-independent coordinatewise decreasing approach for which convergence to the boundary function fails almost everywhere. In contrast, every boundary-point-independent approach whose fixed coordinates increase monotonically to $1$ satisfies an $L^p$ maximal inequality and the almost-everywhere Fatou theorem for $1<p<\infty$.

In this paper, we introduce and study a new class of mappings termed generalized quadrilateral Ciric-Reich-Rus type mappings, which serve as a four-point analogue of Ciric- Reich-Rus type mappings. We establish the existence and non-uniqueness of fixed points, with at most three fixed points possible for these mappings in complete metric spaces under the condition that the mapping has no periodic points of prime period 2 and 3. We determine that while these mappings are often discontinuous within their domain of definition, they must be continuous at their fixed points. By applying additional constraints, notably asymptotic regularity and continuity, we broaden the applicability of fixed-point theorems to include a broader class of mappings. At the end, we attain fixed point theorems for generalized quadrilateral Ciric- Reich-Rus type mappings in metric spaces that are not necessarily complete. Several illustrative examples are provided to validate the theoretical results and demonstrate the independence of our new class from existing notions.

Let a bounded and smooth domain in the n dimensional Euclidean space be given and consider a boundary value problem of the half order Dirac operator and its power, with f be the value of the half derivative of the solution in the domain and, g be the trace value of the solution over the boundary. We will show that the solution will be a generalized function u in the half order Hilbert space which is the orthogonal sum of the solution that emerged as a domain integral from the domain value of the fractional half Dirac operator and the one that is the boundary integrals that emerges from the boundary or trace value of the solution. We use two different symbols to designate orthogonal sums for subspaces and functions in orthogonal subspaces of the Sobolev space.

We show that, for every pair of convex bodies $K,L\subset\mathbb R^n$, $$ \operatorname{vr}(K,L)\leq C\sqrt{n\log(n+1)}. $$ The main point is to place $K$ and $L^\circ$ in isotropic position. We then consider a random orthogonal image of $L$ and control the corresponding operator norm by combining the isotropic mean-gauge estimate of Bizeul and Klartag with Letwin's recent dimension-free bound for the third-moment parameter appearing in their estimate. Our result improves the bound $ \operatorname{vr}(K,L)\leq C\sqrt n \log(n+1)$ proved by Giannopoulos and Hartzoulaki, which had remained the best general estimate for nearly two and a half decades.

Continuing the study of mother-body measures with algebraic Cauchy transform, we associate with a positive algebraic germ $f$, its irreducible equation $P(z,w)=0$, and a compact convex set $K$ the functional \[ \mathfrak D_{P,K}(\mu)= \int_K\left|P\bigl(z,\mathcal C_\mu(z)\bigr)\right|^{1/d}\,dA(z), \qquad d=\text{deg}_wP, \] on the set of positive measures supported in $K$ and whose Cauchy transform has germ $f$ at infinity. We prove continuity of $\mathfrak D_{P,K}(\mu)$ and attainment of its minimum, characterize zero defect, and obtain the estimate \[ \mu(D(a,r))\le C_1r+C_2\mathfrak D_{P,K}(\mu)/r \] away from the zero set of the leading coefficient of $P$, where $D(a,r)$ is the disk of radius $r$ centered at $a$. We establish a dual formula for the defect in the rational case and construct positive algebraic Cauchy transforms by Herglotz theory, Fuss–Catalan and Raney laws, positive sums, polynomial pushforwards, and branch graphs. The passage from zero defect to a mother body is made under the planar-null hypothesis of the support.

For $0<\alpha\leq1$, define $(\mathcal T_\alpha f)(z) :=\int_{B(z,\alpha(1-|z|))}f(\zeta)\,dA(\zeta)$ for $z\in\mathbb D$, where $dA$ is planar Lebesgue measure. We prove that $\mathcal T_\alpha$ is injective on $C(\mathbb D)\cap L^\infty(\mathbb D)$ for $0<\alpha<1$, and that $\mathcal T_1$ is injective on $L^1(\mathbb D)$. In contrast, for each $0<\alpha<1$ there is an injective linear map from $C_c^\infty((0,\alpha))$ into the kernel of $\mathcal T_\alpha$ on $C^\infty(\mathbb D)$; every nonzero function in its image is necessarily unbounded near $\partial\mathbb D$. Under the area-measure interpretation, these results give a complete answer to Hayman–Lingham Problem~7.29, attributed there to L.~Zalcman. The proof combines generalized Abel equations, an Euler–Poisson–Darboux energy argument, and Volterra continuation.

We prove a Bismut-Elworthy-Li formula for linear stochastic evolution equations in Banach spaces. The admissible directions are those for which the deterministic orbit is square integrable with values in the Hilbertian range of the noise coefficient. Under this square function condition, the derivative of the semigroup is represented by a stochastic integral against an explicit deterministic control. The result is a concrete control realisation of the Cameron–Martin formula for the fixed-time Gaussian transition measures.

We establish global $L^p$-Poincaré inequalities, for $ p\in[1,\infty)$, on a class of nondoubling hyperbolic-type metric measure spaces. The proof relies on a discretisation of the space, which gives rise to a Gromov hyperbolic graph, called spiderweb, quasi-isometric to the original space. We prove global Poincaré inequalities for spiderwebs endowed with suitable measures and develop a general transference principle from discrete graphs to metric measure spaces. Combining these results yields global Poincaré inequalities under natural geometric and measure assumptions on the base space.

We establish two pointwise endpoint limits for nonlocal operators: one based on normalized integrals and a distributional limit based on weighted weak-type norms. As applications, the limits yield pointwise Bourgain–Brezis–Mironescu, Maz'ya–Shaposhnikova, Brezis–Seeger–Van Schaftingen–Yung and Gu–Yung formulas, together with higher-order and mean-oscillation variants. We then investigate endpoint limits for fractional powers generated by semigroups and for approximation processes, and derive sharp strong and weak endpoint estimates for operators in harmonic analysis. Finally, while Domínguez and Milman obtained weak-type estimates on product spaces for $p>1$ and left the endpoint $p=1$ open (see [Adv.~Math.~411 (2022), Paper~No.~108774, p.~22]), we prove a sharp pointwise limit in the parameter variable and obtain two–sided iterated weak-type estimates at $p=1$.

Let $X$ be a normed linear space, $Y$ a Banach space, $G\subset X$ an open set and $f:G \to Y$ a mapping. We show that the Fréchet derivative $f'$ of $f$ is of the first Baire class on the (possibly empty) set $D\subset G$ where it is defined.

Let $Z_2$ denote the real Kalton-Peck space. No hyperplane of $Z_2$ admits a complex structure, and $Z_2$ is even in the sense of Ferenczi-Galego. Therefore, $Z_2$ is not linearly isomorphic to its hyperplanes, solving a long-standing conjecture in Banach space theory. The argument uses the Kalton-Swanson symplectic form on $Z_2$, the Castillo-González-Pino Fredholm theorem in $Z_2$, and a special case of a mod-$2$ theorem of Li-Zhu on skew-adjoint Fredholm relations in symplectic spaces, for which a self-contained proof is included.

Let $C_n$ be the optimal constant with the following property. For every even, continuous, strictly positive density $f$ on $R^n$ and all origin-symmetric convex bodies $K,L\subset R^n$, the inequalities $$ \int_{K\cap\xi^\perp}f \leq \int_{L\cap\xi^\perp}f \qquad\text{for all }\xi\in S^{n-1} $$ imply $\int_Kf\leq C_n\int_Lf$. In an earlier paper the authors proved that $C_n\leq\sqrt n$. In this paper, we prove the matching lower bound $C_n\geq c\sqrt n$. To simplify the exposition, we first give a complete one-scale construction, based on earlier work of Klartag and Koldobsky, which yields $C_n\geq c\sqrt{n/\log n}$. For the sharp result, we use the random-rounding construction of Klartag and Livshyts as a black box and combine it with a spherical-averaging support-separation argument.

For a holomorphic vector bundle $F$ of rank $r$ on a smooth Riemann surface $X$ we construct a trace map on the chiral homology of the chiral Clifford algebra $\CE$ attached to the purely odd bundle $E=\Pi(F\oplus F^\vee\otimes\omega_X)$. It is the chiral-algebraic realization of the rank two fermionic vertex operator superalgebra. The free fermion (bc-type) conformal field theory built from a dual pair of odd fields $ \beta_i\in F$, $\gamma^j\in F^\vee\otimes\omega_X$. We give a complete construction of this vertex operator superalgebra, its associated vertex superalgebra bundle, and the isomorphism between the latter's chiral algebra and the chiral envelope $\CE$. Using the Batalin-Vilkovisky (BV) formalism together with Feynman diagrams we prove that the resulting trace map \[ \Trch : \bigl(\widetilde\sC^{\ch}(X,\CE)_{\sQ},\, \dch_{\CE}\bigr)\longrightarrow (\OBV,-\DBV) \] is a chain map satisfying a generalized quantum master equation and is a quasi-isomorphism, generalizing to the odd/Clifford setting the trace map on chiral Weyl algebras constructed by Gui for symplectic bosons. We establish existence, homotopy uniqueness, and functoriality (including metric-independence up to chain homotopy) of the trace map, prove cyclicity of the relevant supertrace, and verify nilpotency of $\DBV$, the graded Leibniz rule, $d^2=0$ for every differential introduced. As an application we compute the trace map on a modified affine current and on a modified energy-momentum tensor, recovering, purely algebraically from the chiral chain complex, Fay's classical formulas for the variation of the fermionic (Ray-Singer) analytic torsion along the moduli of the bundle $F$ and along the moduli of the curve $X$.

In this paper, we show existence and uniqueness of solutions to the viscous Burgers' equation in $\mathbb{R}^d$, when the initial condition $u_0$ is in Hermite-Sobolev space of index $p$, for suitable non-negative integers $p$. Our solutions are local in time. We also have a regularity result, viz. if $u_0$ belongs to Schwartz space, then so does the solution.

In this paper, we obtain the structure of a class of contractive projections, known as generalized tri-circular projections, on some functional Banach spaces on the open unit disk $\mathbb{D}$, which includes Hardy, Bergman and Bloch spaces. We then consider the space $S^p_{\mathcal{K}}$ of $\mathcal{K}$-valued analytic functions on $\mathbb{D}$ such that $f' \in H^p(\mathcal{K})$ and give complete description of generalized tri-circular projections on this space. Here, $\mathcal{K}$ is a complex separable Hilbert space, and $H^p(\mathcal{K})$ denotes the Hardy space of $\mathcal{K}$-valued analytic functions on $\mathbb{D}$.

In this paper, we establish necessary and sufficient conditions for continuous embeddings between weighted local Morrey-type spaces and weighted complementary local Morrey-type spaces. Rather than relying on standard multidimensional approaches, our methodology is based on a one-dimensional reduction. Specifically, we reduce the multidimensional embedding problems to corresponding weighted inequalities involving one-dimensional Cesàro and Copson function spaces. Applying this reduction alongside recent results in the characterization of the embeddings between Cesàro and Copson spaces, we provide complete characterizations for the embeddings \[ \LM_{p_1,q_1}(v_1, w_1) \hookrightarrow \LM_{p_2,q_2}(v_2, w_2)\] and \[\dual\LM_{p_1,q_1}(v_1, w_1) \hookrightarrow \LM_{p_2,q_2}(v_2, w_2).\] Our results cover all possible finite cases of the parameters $0 < p_1, p_2, q_1, q_2 < \infty$.

Given a growth function $\varphi:[0,\infty)\rightarrow [0,\infty)$, it is established that the commutators generated by sublinear operators and Orlicz-$\mathrm{BMO}$ function $b$ are bounded from $H_{b}^{\varphi}(\mathbb{R}^{n})$ to $L^{1}(\mathbb{R}^{n})$, and from $H^{\varphi}(\mathbb{R}^{n})$ to $L^{1,\,\infty}(\rn)$, where $H_{b}^{\varphi}(\mathbb{R}^{n})$ is a specific subspace of Orlicz-Hardy space $H^{\varphi}(\mathbb{R}^{n})$ and sublinear operators include Lusin area integral, g-function, Marcinkiewicz integral and Bochner-Riesz mean operator. Under the assumptions $T^*1=0$ and $T^*b=0$, it is shown that the Orlicz-$\mathrm{BMO}$ commutator associated with the Bochner-Riesz mean operator admits endpoint boundedness from $H_{b}^{\varphi}(\mathbb{R}^{n})$ to $H^{1}(\mathbb{R}^{n})$. However, the commutators corresponding to other operators discussed in this paper do not possess the aforementioned endpoint boundedness, and a counterexample is provided to illustrate this point.