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For $m,n\in\mathbb{N}$ let $X=(X_{ij})_{i\leq m,j\leq n}$ be a random matrix, $A=(a_{ij})_{i\leq m,j\leq n}$ a real deterministic matrix, and $X_A=(a_{ij}X_{ij})_{i\leq m,j\leq n}$ the corresponding structured random matrix. We study the expected operator norm of $X_A$ considered as a random operator between $\ell_p^n$ and $\ell_q^m$ for $1\leq p,q \leq \infty$. We prove optimal bounds up to logarithmic terms when the underlying random matrix $X$ has i.i.d. Gaussian entries, independent mean-zero bounded entries, or independent mean-zero $\psi_r$ ($r\in(0,2]$) entries. In certain cases, we determine the precise order of the expected norm up to constants. Our results are expressed through a sum of operator norms of Hadamard products $A\circ A$ and $(A\circ A)^T$., Comment: 50 pages, 1 figure, 1 table; Remark 1.1 and Subsection 5.4 added, typos corrected

Let $0

We prove that in the context of general Markov semigroups Beckner inequalities with constants separated from zero as $p\to 1^+$ are equivalent to the modified log Sobolev inequality (previously only one implication was known to hold in this generality). Further, by adapting an argument by Boucheron et al. we derive Sobolev type moment estimates which hold under these functional inequalities. We illustrate our results with applications to concentration of measure estimates (also of higher order, beyond the case of Lipschitz functions) for various stochastic models, including random permutations, zero-range processes, strong Rayleigh measures, exponential random graphs, and geometric functionals on the Poisson path space., Comment: 56 pages, 1 figure; presentation in Sec. 4.7 changed

Probability measures satisfying a Poincar{\'e} inequality are known to enjoy a dimension free concentration inequality with exponential rate. A celebrated result of Bobkov and Ledoux shows that a Poincar{\'e} inequality automatically implies a modified logarithmic Sobolev inequality. As a consequence the Poincar{\'e} inequality ensures a stronger dimension free concentration property , known as two-level concentration. We show that a similar phenomenon occurs for the Latala-Oleszkiewicz inequalities, which were devised to uncover dimension free concentration with rate between exponential and Gaussian. Motivated by the search for counterexamples to related questions, we also develop analytic techniques to study functional inequalities for probability measures on the line with wild potentials.

Let $H$ be the Hardy operator and $I$ the identity operator acting on functions on the real half-line. We find optimal bounds for the operator $H - I$ in the setting of power weights and the cases of positive decreasing functions, positive functions, and general functions. As a byproduct, we obtain some results about the optimal relations between the norms of $H$ and its dual., Comment: 22 pages

We present precise multilevel exponential concentration inequalities for polynomials in Ising models satisfying the Dobrushin condition. The estimates have the same form as two-sided tail estimates for polynomials in Gaussian variables due to Lata{\l}a. In particular, for quadratic forms we obtain a Hanson-Wright type inequality. We also prove concentration results for convex functions and estimates for nonnegative definite quadratic forms, analogous as for quadratic forms in i.i.d. Rademacher variables, for more general random vectors satisfying the approximate tensorization property for entropy.

We prove that for a probability measure on $\mathbb{R}^n$, the Poincar\'e inequality for convex functions is equivalent to the weak transportation inequality with a quadratic-linear cost. This generalizes recent results by Gozlan et al. and Feldheim et al., concerning probability measures on the real line. The proof relies on modified logarithmic Sobolev inequalities of Bobkov-Ledoux type for convex and concave functions, which are of independent interest. We also present refined concentration inequalities for general (not necessarily Lipschitz) convex functions, complementing recent results by Bobkov, Nayar and Tetali., Comment: v2: 26 pages, Remark 6.8. slightly modified, Theorem 7.2. added

We show that every symmetric random variable with log-concave tails satisfies the convex infimum convolution inequality with an optimal cost function (up to scaling). As a result, we obtain nearly optimal comparison of weak and strong moments for symmetric random vectors with independent coordinates with log-concave tails., Comment: 11 pages

We give a sufficient and necessary condition for a probability measure $\mu$ on the real line to satisfy the logarithmic Sobolev inequality for convex functions. The condition is expressed in terms of the unique left-continuous and non-decreasing map transporting the symmetric exponential measure onto $\mu$. The main tool in the proof is the theory of weak transport costs. As a consequence, we obtain dimension-free concentration bounds for the lower and upper tails of convex functions of independent random variables which satisfy the convex log-Sobolev inequality., Comment: 17 pages

We calculate the norms of the operators connected to the action of the Beurling-Ahlfors transform on radial function subspaces introduced by Ba\~nuelos and Janakiraman. In particular, we find the norm of the Beurling-Ahlfors transform acting on radial functions for $p>2$, extending the results obtained by Ba\~nuelos and Janakiraman, Ba\~nuelos and Os\c{e}kowski, and Volberg for $1

We provide a mild sufficient condition for a probability measure on the real line to satisfy a modified log-Sobolev inequality for convex functions, interpolating between the classical log-Sobolev inequality and a Bobkov-Ledoux type inequality. As a consequence we obtain dimension-free two-level concentration results for convex function of independent random variables with sufficiently regular tail decay. We also provide a link between modified log-Sobolev inequalities for convex functions and weak transport-entropy inequalities, complementing recent work by Gozlan, Roberto, Samson, and Tetali., Comment: 25 pages; changes: references and comments about recent results by other Authors added, hypercontractive estimates in Section 3 added, a few typos corrected; accepted for publication in Studia Mathematica

Let $\mathcal{H}^{\mathbb{T}}$ denote the Hilbert transform on the circle. The paper contains the proofs of the sharp estimates \begin{equation*} \frac{1}{2\pi}|\{ \xi\in\mathbb{T} : \mathcal{H}^{\mathbb{T}}f(\xi) \geq 1 \}| \leq \frac{4}{\pi}\arctan\left(\exp\left(\frac{\pi}{2}\|f\|_1\right)\right) -1, \quad f\in L^{1}(\mathbb{T}), \end{equation*} and \begin{equation*} \frac{1}{2\pi}|\{ \xi\in\mathbb{T} : \mathcal{H}^{\mathbb{T}}f(\xi) \geq 1 \}| \leq \frac{\|f\|_2^2}{1+\|f\|_2^2},\quad f\in L^{2}(\mathbb{T}). \end{equation*} Related estimates for orthogonal martingales satisfying a subordination condition are also established., Comment: 10 pages