Papers

We study the projection outer volume ratio ${\rm povr}_k(K)$ introduced by D. Galicer, A.E. Litvak, M. Merzbacher, and D. Pinasco. When $k\leq n/2$, we provide sharp bounds in the general case, namely \[ {\rm povr}_k(K)\leq C \max\Big\{1, \sqrt{\frac{k}{\ln(2n/k)}}\Big\}, \] where the maximal order is obtained for $K=B_1^n$. Moreover, in the important particular case $K=B_2^n$ we show that ${\rm povr}_k(B_2^n)$ is bounded by an absolute constant whenever $k\leq n/2$ and \[ {\rm povr}_k(B_2^n)\leq C\sqrt{\frac{n}{n-k}}\, \ln\left(1+\frac{n}{n-k}\right) \] for $k>n/2$. The latter bound is sharp up to a logarithmic term.

We formulate the shortest-path problem in a domain containing a crack as a Hamilton–Jacobi equation carrying a state constraint on the crack, which on the binding part of the crack takes the form of a variational inequality: the crack is a two-sided barrier that may be touched and followed but not traversed, so that it acts as a lower-dimensional state constraint, carried separately by each of its two lips, which admits tangential propagation but forbids crossing. We establish a comparison principle and uniqueness of the value function through trajectory-based arguments that circumvent the failure of the classical doubling technique of Crandall, Ishii, and Lions at the crack. The variational inequality induces a free boundary on the crack separating a sliding regime, where optimal trajectories run along the crack over a finite segment, from an illuminated regime, where the crack is reached transversally by direct rays; we give a complete geometric characterization of both regimes, including grazing (tangential) contact, the absence of reflection at regular crack points, and a diffraction law at crack tips, all derived from optimality alone. A simulation of the forward problem on an exactly solvable configuration reproduces the three regimes and the tip-diffraction law to the accuracy of the closed-form solution.

The paper contains the study of the weighted $L^{p_1}\times L^{p_2}\times \ldots \times L^{p_m}\to L^p$ estimates for the multilinear maximal operator, in the context of abstract probability spaces equipped with a tree-like structure. Using the Bellman function method, we identify the associated optimal constants for weights satisfying the multilinear Muckenhoupt condition.

We prove a single-logarithm bound for the eigenvalues in the plunge region of Fourier concentration operators, removing the extra logarithmic factor in the Kulikov–Dam Larsen bound for two general admissible sets throughout each fixed exponential window. For bounded spatial and frequency sets $A,B\subset\mathbb{R}^d$ of positive measure with finite upper codimension-one Minkowski boundary content, the number $\Lambda_\varepsilon(cA,B)$ of eigenvalues in $(\varepsilon,1-\varepsilon)$ satisfies $\Lambda_\varepsilon(cA,B)\le C(d,A,B,\beta)c^{d-1}D(1+\log_+(c/D))$, where $D=\log(1/(\varepsilon(1-\varepsilon)))$, for every fixed $\beta>0$, $c\ge2$, $0<\varepsilon<1/2$, and $D\le\beta c$, with $\log_+t=\max(0,\log t)$. The argument uses positive band-limited smoothing and requires no symmetry, convexity, or connectedness. In dimension two, the corresponding exponential-window estimate has a constant uniform in the window parameter. We also determine the optimal coordinate-Lipschitz constant $d-1$ for recursive decompositions of balls and standard simplices in every dimension, and obtain box-frequency estimates with explicit geometric constants. For $C^1$ convex bodies in three dimensions, the optimal infimum of recursive coordinate slopes is $2$. Uniform real-order Bessel estimates give independent disk and ball bounds at every spectral threshold, including an explicit deep-tail term. For smooth strictly convex planar frequency bodies with positive curvature, we prove correlation estimates for aligned Gaussian packets at arbitrary relative scales and quantitative lower bounds on the cost of estimating spatial blocks separately. Together these results connect the global spectral bound with sharp geometric constructions and the cancellation retained by grouping boundary interactions.

We study continuous embeddings from two classes of generalized weighted Morrey spaces into generalized Morrey spaces. The Komori–Shirai-type spaces use the weighted measure of cubes in their normalization, whereas the Samko-type spaces use the Lebesgue measure. We first consider generalized Morrey spaces on bounded domains. In the non-radial setting, we establish embedding results under local radial comparability assumptions on the Morrey functions. As a consequence, we characterize the corresponding embeddings in the radial setting for the full range $0<p_1,p_2<\infty$. We then study embeddings from generalized Komori–Shirai-type weighted Morrey spaces associated with piecewise power weights and general Muckenhoupt weights. For the Samko-type scale, we treat both generalized and classical weighted Morrey spaces, again considering piecewise power weights and general Muckenhoupt weights. Finally, assuming that the weight is comparable to a positive constant on some cube or ball, we show that the continuous embeddings under consideration are not compact.

The discriminant of a square matrix is defined as a discriminant of its characteristic polynomial. In this paper we provide a description of linear maps preserving the matrix discriminant assuming that the ground field is large enough. Our description generalizes the known results to the case where the field may have a finite cardinality and its characteristic may divide the size of considered matrices. To obtain this description, we extend the approach used in our previous work on the subject and, in addition to that, introduce new invariants for the action of all bijective linear maps preserving a given polynomial, which are on independent interest and could be applied to other problems in the theory of linear preservers.

We give two explicit reformulations of a Balazard–Saias–Yor type criterion for the Riemann hypothesis. First, Jensen's formula is applied to $u(s)=(s-1)\zeta(s)$ after mapping the unit disk onto $Re s<1/2$. For the normalization whose center corresponds to $s=-1/2$, the images $z_n=(4n-1)/(4n+3)$ of the trivial zeros yield a telescoping Jensen product. This isolates, at finite radius, the complete divergent contribution of the trivial zeros; the remaining defect is a nonnegative integrated count of mapped nontrivial zeros off the critical line. A one-parameter version gives a gamma-quotient product and an explicit finite part. Second, after passing through the established boundary identity, we combine two Poisson weights to obtain an absolutely convergent logarithmic integral of the Riemann $\Xi$-function. From the random variable $Z=|\Xi(T)/\Xi(0)|^2$ we define its law $\mu$ and the size-biased law $\nu(\,\mathrm d z)=z\mu(\,\mathrm d z)/M$. The data-processing inequality is lossless for this statistic, and the Riemann hypothesis is equivalent to saturation of an explicit upper bound for $D_{\rm KL}(\mu\|\nu)$. We also record the associated Donsker–Varadhan and moment forms and express the target constant through Riemann's positive Fourier kernel. The analytic boundary criterion and Jensen–Hardy mechanism used here are known. The purpose of the paper is an explicit finite-radius treatment of the trivial zeros and an information-theoretic reformulation, not a new proof of the Riemann hypothesis or an independent boundary-integral criterion.

A conference signal is a complex-valued function on a finite abelian group that vanishes at 0 in both the time and frequency domains, and is otherwise flat in both domains. For example, the Legendre symbol is a conference signal on $\mathbb{Z}_p$ since it vanishes at $0$, takes values $\pm 1$ away from zero, and is a scalar multiple of its Fourier transform. More generally, any nontrivial multiplicative character on a finite field is a conference signal on its additive group. Until now, conference signals have mainly been studied by number theorists, who wanted to know if multiplicative characters were the only conference signals on finite fields. The state of the art uses fancy tools from symplectic geometry to prove that non-character conference signals exist on non-small fields. The same tools show conference signals exist on every abelian group of odd order, without saying what they are. In this (human-friendly!) paper, we explain how conference signals produce other objects of interest in algebraic combinatorics and applied harmonic analysis, and then we present some new, explicit constructions of conference signals on $\mathbb{Z}_n$. In particular, we give the first explicit conference signals for composite $n$ and the first explicit non-character examples for prime $n$.

In the context of locally convex vector valued integrable tensors on open sets of a $n$-dimensional manifold $M$ developped in our previous work \cite{sil09261621}, in this paper we construct a framework in which to address both a variational problem associated with a Lagrangian $\mathscr{L}$, and a conservation law on-shell associated with actions $\phi$ and $\omega$. $\mathscr{L}$ is a $C^{p}$-map valued in a quasi-complete Hausdorff lcs $\mathcal{Z}$ and defined on an open set $O$ of a normed space, $\phi$ is a $C^{p}$-action parametrized by an open $A$ of a normed space $T$, acting over the space $\mathfrak{Z}^{O}$ of fields valued in a normed space $Y_{0}$ and defined on $W$ domain of a chart $\xi$ of $M$; $\omega$ is a $C^{p}$-action on $W$ parametrized by $A$. Via a result in \cite{sil09261621}, we construct $\mathfrak{A}:\mathfrak{Z}^{O}\times A\to\mathcal{M}(W,\mathcal{Z})$ where $\mathfrak{A}(u,\eta)$ is the action functional associated with $\mathscr{L}$, $u$ and $\eta$. Here $\mathcal{M}(W,\mathcal{Z})=\mathcal{L}_{s}(\mathcal{H}(W),\mathcal{Z})$ is the lcs of measures on $W$ and valued in $\mathcal{Z}$. By letting $\mathcal{A}(u):A\to\mathcal{M}(W,\mathcal{Z})$ be such that $\mathcal{A}(u)(\eta)=\mathfrak{A}(u,\eta)$, we state that $\mathcal{A}(u)$ is differentiable off-shell, find its differential map off and on-shell. We establish that if the differential map of $\mathcal{A}(u)$ on $0$ is $0$, then $\mathscr{N}[u]$ is conserved on-shell, and if $p=\infty$, then the divergence of $\mathscr{N}[u]$ equals $0$, where the divergence of a lcs valued vector field is defined and analyzed in \cite{sil09261621}. Here $\mathscr{N}[u]$ is a $\mathcal{L}_{b}(T,\mathcal{Z})$-valued $C^{(p-1)}$-vector field of $M$ on $U=\xi(W)$.

We prove that there exists an absolute constant $c>1$ such that for every $n \geq 3$ there exists an $n$-dimensional real normed space $X$ for which every linear projection $P$ onto a hyperplane satisfies $\|P\| \geq c$. More generally, for every integer $k \geq 1$ there exists a constant $c_k>1$ such that for every $n \geq k+2$ there exists an $n$-dimensional real normed space $X$ such that every projection $P$ onto a subspace of codimension at most $k$ satisfies $\|P\| \geq c_k$. The proof is based on an idea of Gleit and Lazar, who established an analogous result for the space of real $n \times n$ matrices equipped with either the trace-class norm or the operator norm

A classical phenomenon distinguishing infinite-dimensional holomorphy from its finite-dimensional counterpart is that monomial expansions of holomorphic functions need not converge absolutely. In this paper, we systematically investigate monomial convergence on complex Banach sequence spaces. We establish general structural results and give a negative answer to Lempert's problem. We then determine the domains of absolute monomial convergence for holomorphic functions on the classical Banach sequence spaces and obtain a complete classification for continuous homogeneous polynomials. As a further application, we establish a sharp comparison between two notions of convergence radius for homogeneous Taylor expansions on $\ell^1$. The methods developed here are of independent interest and may be applied in other places.

We study Hamiltonian elliptic systems with prescribed bilinear mass $\int_{\mathbb{R}^N}uv=a>0$. We establish a Gagliardo–Nirenberg inequality in the crossed gradient pairing and bilinear mass, on scaling-invariant component-bounded classes. It identifies the critical curve $1/p+1/q=N/(N+2)$ and the restricted energy trichotomy. For power nonlinearities, we establish uniqueness up to common translations and nondegeneracy of positive profiles throughout the Sobolev-subcritical hyperbola. Exact scaling then classifies positive normalised solutions and identifies the unique critical mass. In dimension two, we develop a bilinear exponential Gagliardo–Nirenberg estimate with a sharp gradient threshold and a truncated scalar refinement with the optimal quartic leading coefficient. Adapting variational ideas from [Cassani-Tarsi, Calc.Var.PDE (2015)] we combine a mass-normalising quotient, reduction over the full negative fibres and compactness below the concentration threshold to obtain least-energy positive radial solutions for pure exponential nonlinearities at every prescribed mass below the cubic limiting mass. Local perturbative branches and a small-frequency exponential branch yield explicit mass-response formulas.

We establish an Alexandrov–Fenchel type inequality in which one of the reference bodies is replaced by a function whose Hessian satisfies a balancing condition on its eigenvalues, without being required to be positive semidefinite. The proof hinges on a new extension of Alexandrov's inequality for mixed discriminants. As applications, we derive a log-concavity principle for linear functionals of area measures, answering a question of Colesanti, Hug, and Saorín-Gómez, as well as Brunn–Minkowski type inequalities for intrinsic volumes of mean section bodies, answering a question of Schuster.

A Banach space $X$ is $L$-embedded in its bidual if there is a projection $P\colon X^{**}\to X$ such that $\|x\|=\|Px\|+\|x-Px\|$ for all $x\in X^{**}$. We show that, as long as $X$ has dimension at least $2$, its Lipschitz-free space, denoted by $\mathcal{F}(X)$, is not $L$-embedded in its bidual. Our methods have applications to the problem of when the $L$-embeddability of $\mathcal{F}(M)$ passes to $\mathcal{F}(A)$ for a metric space $M$ and $A\subseteq M$. This is the case when $A$ is compact or when $A$ is geodesically closed.

Although generalized hyperbolicity has proved to be a fundamental notion in linear dynamics, recent results show that it is too restrictive to characterize the shadowing property of invertible bounded linear operators on Banach spaces. This motivates the introduction of generalized pseudo-hyperbolicity, a weaker notion formulated through continuous homogeneous maps generating exponentially decaying Green families. We prove that, for arbitrary sequences of invertible bounded linear operators on Banach spaces, generalized pseudo-hyperbolicity is equivalent to both shadowing and topological stability. In the autonomous setting, this yields the equivalence between shadowing and topological stability, removing additional assumptions from previous results. We also prove robustness of shadowing and topological stability for sequences of invertible bounded linear operators.

In this paper, we construct the Mehler-Fock transform of arbitrary order MF(mu,k) by removing the restriction Re(mu) > -1/2 with the help of a positive integer k such that Re(mu+k) > -1/2 and obtain some transformation formulae based on the differential operator M(mu)N(mu). Furthermore, we define the Mehler-Fock translation and convolution operators of arbitrary order and demonstrate that the factorization property for the convolution holds in the test function space H(mu).

We study the Dunkl–Schrödinger operator $L=-\Delta_k+V$ on $\mathbb R^N$, $N>2$, where $V$ is a nonnegative potential in the reverse Hölder class ${\rm RH}^q(dw)$, $q>\mathbf N/2$, and $\mathbf N$ is the homogeneous dimension. We establish decay estimates for the resolvent kernel in terms of the critical radius and the orbit distance. For reflection-invariant potentials, we obtain $L^p(dw)$ bounds for $VL^{-1}$, second-order Dunkl derivatives of $L^{-1}$, and the Riesz transform $\nabla_kL^{-1/2}$, together with fractional potential and mixed estimates. The proofs combine local estimates for weak solutions, the Dunkl Fefferman–Phong inequality, and kernel estimates adapted to the Euclidean and orbit distances.

We develop a theory of distributional Riesz fractional gradients in variable exponent Lebesgue spaces and introduce the associated nonlocal Sobolev spaces. Moreover, Sobolev, Hardy, Morrey, and Trudinger-type inequalities are established, and a fractional fundamental theorem of calculus is derived. As an application, we study nonlinear variational problems with nonstandard growth and establish existence and Euler–Lagrange equations for minimizers.

We develop a sparse domination framework adapted to nested families in general measure spaces, possibly including point masses. We show that the Carleson condition is equivalent to a slight generalization of the usual sparse condition, and use this characterization to obtain one- and two-weight estimates for the associated averaging operators, together with converse testing conditions. We apply our general results to integral operators on locally finite trees. Trees come naturally equipped with two nested geometries: sectors and pruned sectors. We prove pointwise sparse domination of natural versions of the harmonic Bergman projector by the corresponding averaging operators, depending on whether the measure on the tree is doubling or not, and show that the distinction is necessary in general. For the positive Bergman projector, we obtain pointwise comparability. As a consequence, we obtain new weighted inequalities and boundedness results for the Bergman projector and a class of Toeplitz-like multipliers.

We study C*-algebras generated by representations of product systems over unital subsemigroups of amenable discrete groups. In the first part, we show that exactness of the generated C*-algebra is equivalent to exactness of the cores over the constructible ideals. Nuclearity of the generated C*-algebra is equivalent to nuclear embeddability of the cores in the fixed point algebra. In the second part, we further study injective equivariant Fock covariant representations, and whether the properties of the cores are inherited from properties of the coefficient C*-algebra, as in the case of compactly aligned saturated product systems over right LCM semigroups. If the semigroup is finitely aligned and the product system is saturated and compactly aligned, then exactness of the cores is equivalent to exactness of the coefficient C*-algebra. If in addition the semigroup is strongly finitely aligned, then nuclear embeddability of the cores is equivalent to nuclear embeddability of the coefficient C*-algebra. These results do not hold without the compact alignment hypothesis, even for right LCM semigroups.

We study weighted coefficient estimates for the Laguerre transform in terms of moduli of smoothness generated by the Laguerre translation. For every integer $r\ge1$ and $1\le p\le2$, a weighted $\ell^{p'}$ norm of $\min\{1,nt\}^{r}\widehat f_\alpha(n)$ is bounded by the modulus of order $r$; for $2\le p\le\infty$ the reverse direction holds on the same weighted sequence scale. At $p=2$ these estimates yield a two-sided equivalence, and testing individual Laguerre polynomials proves optimality of the multiplier weight up to constants. The proof combines Hausdorff–Young inequalities with an averaged lower bound for the translation multiplier. As consequences we recover Jackson- and Bernstein-type estimates and the known $L^2$ approximation characterisation of the Lipschitz classes for $0<\beta<r$. We also give the corresponding one-sided coefficient-tail criteria for general $p$, together with saturation and endpoint statements.

We study the Cauchy problem for a semilinear heat equation driven by the fractional harmonic oscillator. We establish local and global well-posedness in Orlicz and Morse–Transue spaces, based on sharp semigroup estimates and an integral compatibility condition between the Young function and the nonlinearity. We identify a critical Young function separating local well-posedness from instantaneous nonexistence for suitable data. Finally, we prove a confinement effect: under a Dini condition at the origin, small data yield global solutions decaying at the sharp exponential rate, with no Fujita-type restriction, in contrast with the corresponding free equation.

In this paper, we investigate the differentiability of Korevaar–Schoen $p$-energy forms and the associated $p$-energy measures. We obtain higher-order derivatives by virtue of explicit realizations of Korevaar–Schoen $p$-energy forms and the associated $p$-energy measures as subsequential pointwise limits of certain double integrals.

The Kannan–Lovász–Simonovits (KLS) conjecture asserts that isotropic log-concave probability measures have Poincaré constants bounded by a universal constant, independently of dimension. We give a deterministic variational proof with an explicit bound on the Poincaré constant $C_P(\mu)\le25$, where $\mu$ is any isotropic log-concave probability measure. Starting from elliptic moment estimates and a quadratic variance inequality, we establish geometric bounds on normalized Appell coefficient norms through joint maximization over the measure and test function. Variation of the measure gives a maximum-principle inequality, while stationarity in the test function controls the highest-order cumulant terms. Two concave barriers constructed from quadratic and cubic polynomials close the induction. A weighted Helmholtz–Hodge decomposition then controls the curl correction of weighted divergence, yielding a curvature estimate for compatible symmetric tensor fields that is uniform in rank. Quadratic duality converts this estimate into an operator comparison for centered integration, linking the coefficient bounds to control of the inverse gradient. A spectral-radius estimate and a scalar growth inequality for adjoint iterates then yield the Poincaré bound. The final estimate is independent of the auxiliary positive curvature, allowing approximation to complete the proof for general isotropic log-concave measures.

We study sampling discretization of the uniform norm for trigonometric polynomials with frequencies in a hyperbolic cross of level $N$ on $\T^d$. For every $d,N\ge2$ and $\varepsilon\in(0,1)$, we construct an explicit norming set with at most $\bigl(C_1(1+\varepsilon^{-1})\bigr)^{d-1}N^{1+\varepsilon}$ points and norming constant at most $\bigl(C_2(1+\varepsilon^{-1})\bigr)^{d-1}$, where $C_1,C_2$ are absolute constants. We also prove that independent Haar-distributed points achieve the same exponent $1+\varepsilon$: a sample of size at least $C(d,\varepsilon)N^{1+\varepsilon}\log(2/\eta)$ norms the entire space simultaneously with probability at least $1-\eta$, with a norming constant bounded by $\exp(C_d\varepsilon^{-d})$ and independent of $N$. The deterministic construction combines uniformly stable de la Vallée Poussin sampling operators with a Smolyak-type combination identity. The random result follows from an abstract norming theorem for sums of spaces associated with commuting regular partitions, together with a multiscale approximation of hyperbolic-cross polynomials. Finally, we show that every norming set of constant $B>1$ for the trigonometric polynomials with frequencies in $\{-1,0,1\}^d$ has at least $\bigl(\pi d/(2e\log B)\bigr)^{d/2}$ points. This rules out quadratic bounds in the dimension of the polynomial space with a prefactor growing only polynomially in $d$ when the norming constant is fixed.

Let $\varepsilon_1,\ldots,\varepsilon_n$ be independent Rademacher random variables. We settle the best constant $c$ such that \[ \mathbb{E}\Bigl|\sum_{j=1}^n a_j\varepsilon_j\Bigr| \ge \frac{1}{\sqrt{2}} +c\Bigl|a-\frac{e_1+e_2}{\sqrt{2}}\Bigr| \] holds for every $n\ge 2$ and every unit vector $a\in\mathbb{R}^n$ with $a_1\ge\cdots\ge a_n\ge 0$. We prove that the optimal constant is $c=(2-\sqrt{2})^{3/2}/8$, with equality attained at $a=\frac{1}{2}(1,1,1,1,0,\ldots,0)$ for $n\ge 4$. This completes a line of research by De-Diakonikolas-Servedio, Eskenazis-Nayar-Tkocz and Fang-Wang.

A classical theorem of Figiel asserts that every isometric embedding $\varphi\colon X\to Y$ between real Banach spaces, with $\varphi(0)=0$, admits a linear contractive left inverse on the closed linear span of $\varphi(X)$. We show that this phenomenon breaks down completely once local convexity is lost. More precisely, for every $0<p<1$ and every nonzero real Banach space $X$, we construct a $p$-Banach space $Y$ and an isometric embedding $\varphi\colon X\to Y$ such that $\varphi(0)=0$ and the closed linear span of $\varphi(X)$ is the entire space $Y$, but there is no linear map, bounded or not, on the linear span of $\varphi(X)$ which is a left inverse of $\varphi$. The construction is carried out using the $p$-Banach counterpart of Lipschitz-free spaces. In fact, $Y$ may be chosen isomorphic to the Lipschitz-free $p$-space over $X$. The main ingredient is a quantitative estimate controlling the distance to elementary molecules in Lipschitz-free $p$-spaces, a phenomenon specific to the range $0<p<1$. As a further consequence, we show that a real Banach space has the isometric $p$-lifting property if and only if its dimension is at most one.

We construct a Banach lattice $d$-algebra with a strong order unit whose product has no positive associative bilinear extension to its Dedekind completion. This gives a negative answer to a question of Wickstead and to the $d$-algebra case of a question of Huijsmans. It also disproves an extension theorem of Chil. The example shows that norm completeness and a strong order unit do not ensure that multiplication extends to the Dedekind completion.

We solve two long outstanding problems here. We first solve in ZFC a problem that the first author and Weaver posed ten years ago: a state on a von Neumann algebra is countably additive on orthogonal projections if and only if it is sequentially weak* continuous, equivalently, sequentially normal. Such states may be considered the quantum (countably additive) probability measures, or rather their extension to a noncommutative integral. The principal tools are known ideas from direct integral theory from the 1970s. Indeed we prove a disintegration theorem for states on products of sigma-finite von Neumann algebras. We give several applications. For example we give a new variant of Gleason theorem, extending countably additive probability measures on projections to sequentially weak* continuous states when the algebra has no type $I_2$ direct summand. We also establish normality and sequential normality criteria for weights, and give several other applications of our main result to weights. In forthcoming work we consider many applications to `quantum measure and integration theory', such as variants of Lebesgue's dominated convergence theorem for von Neumann algebras. The analytic tools and product-quotient construction developed here are used to prove in an Appendix by the first author that normality and sequential normality of arbitrary weights are determined by their restrictions to abelian von Neumann subalgebras, resolving Haagerup's Problem 1.11. (This may be moved to a companion paper.) That is, normality of weights is commutatively determined.

We establish an $L^1$-theory for a class of elliptic operators with unbounded coefficients, extending our earlier semigroup generation results and those of Canale–Rhandi–Tacelli to the endpoint $p=1$. The $L^1$-realization generates a positive analytic $C_0$-semigroup that is compact for every positive time. We characterize the generator domain through weighted estimates and prove consistency with the corresponding $L^p$-semigroups for $1<p<\infty$.