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Register a new userNonhomogeneous $T(1)$ Theorem on Product Quasimetric Spaces
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In this paper, we provide a non-homogeneous $T(1)$ theorem on product spaces $(X_1 times X_2, rho_1 times rho_2, mu_1 times mu_2)$ equipped with a quasimetric $rho_1 times rho_2$ and a Borel measure $mu_1 times mu_2$, which, need not be doubling but satisfies an upper control on the size of quasiballs.
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In this paper, we prove a $Tb$ theorem on product spaces $Bbb R^ntimes Bbb R^m$, where $b(x_1,x_2)=b_1(x_1)b_2(x_2)$, $b_1$ and $b_2$ are para-accretive functions on $Bbb R^n$ and $Bbb R^m$, respectively.
Let $A$ be a compact set in $mathbb{R}$, and $E=A^dsubset mathbb{R}^d$. We know from the Mattila-Sjolins theorem if $dim_H(A)>frac{d+1}{2d}$, then the distance set $Delta(E)$ has non-empty interior. In this paper, we show that the threshold $frac{d+1}{2d}$ can be improved whenever $dge 5$.
By a result of Schur [J. Reine Angew. Math. 1911], the entrywise product $M circ N$ of two positive semidefinite matrices $M,N$ is again positive. Vybiral [Adv. Math. 2020] improved on this by showing the uniform lower bound $M circ overline{M} geq E_n / n$ for all $n times n$ real or complex correlation matrices $M$, where $E_n$ is the all-ones matrix. This was applied to settle a conjecture of Novak [J. Complexity 1999] and to positive definite functions on groups. Vybiral (in his original preprint) asked if one can obtain similar uniform lower bounds for higher entrywise powers of $M$, or for $M circ N$ when $N eq M, overline{M}$. A natural third question is to obtain a tighter lower bound that need not vanish as $n to infty$, i.e. over infinite-dimensional Hilbert spaces. In this note, we affirmatively answer all three questions by extending and refining Vybirals result to lower-bound $M circ N$, for arbitrary complex positive semidefinite matrices $M, N$. Specifically: we provide tight lower bounds, improving on Vybirals bounds. Second, our proof is conceptual (and self-contained), providing a natural interpretation of these improved bounds via tracial Cauchy-Schwarz inequalities. Third, we extend our tight lower bounds to Hilbert-Schmidt operators. As an application, we settle Open Problem 1 of Hinrichs-Krieg-Novak-Vybiral [J. Complexity, in press], which yields improvements in the error bounds in certain tensor product (integration) problems.
In this article, we extend a result of L. Loomis and W. Rudin, regarding boundary behavior of positive harmonic functions on the upper half space $R_+^{n+1}$. We show that similar results remain valid for more general approximate identities. We apply this result to prove a result regarding boundary behavior of nonnegative eigenfunctions of the Laplace-Beltrami operator on real hyperbolic space $mathbb H^n$. We shall also prove a generalization of a result regarding large time behavior of solution of the heat equation proved in cite{Re}. We use this result to prove a result regarding asymptotic behavior of certain eigenfunctions of the Laplace-Beltrami operator on real hyperbolic space $mathbb H^n$.
In this paper we give a geometric condition which ensures that $(q,p)$-Poincare-Sobolev inequalities are implied from generalized $(1,1)$-Poincare inequalities related to $L^1$ norms in the context of product spaces. The concept of eccentricity plays a central role in the paper. We provide several $(1,1)$-Poincare type inequalities adapted to different geometries and then show that our selfimproving method can be applied to obtain special interesting Poincare-Sobolev estimates. Among other results, we prove that for each rectangle $R$ of the form $R=I_1times I_2 subset mathbb{R}^{n}$ where $I_1subset mathbb{R}^{n_1}$ and $I_2subset mathbb{R}^{n_2}$ are cubes with sides parallel to the coordinate axes, we have that % begin{equation*} left( frac{1}{w(R)}int_{ R } |f -f_{R}|^{p_{delta,w}^*} ,wdxright)^{frac{1}{p_{delta,w}^*}} leq c,delta^{frac1p}(1-delta)^{frac1p},[w]_{A_{1,mathfrak{R}}}^{frac1p}, Big(a_1(R)+a_2(R)Big), end{equation*} % where $delta in (0,1)$, $w in A_{1,mathfrak{R}}$, $frac{1}{p} -frac{1}{ p_{delta,w}^* }= frac{delta}{n} , frac{1}{1+log [w]_{A_{1,mathfrak{R}}}}$ and $a_i(R)$ are bilinear analog of the fractional Sobolev seminorms $[u]_{W^{delta,p}(Q)}$ (See Theorem 2.18). This is a biparameter weighted version of the celebrated fractional Poincare-Sobolev estimates with the gain $delta^{frac1p}(1-delta)^{frac1p}$.
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