arXiv subjects
Publications and source records attributed to Dmitriy Stolyarov.
Inspired by a conjecture of Vladimir Maz'ya on $Φ$-inequalities in the spirit of Bourgain and Brezis, we establish some $Φ$-inequalities for fractional martingale transforms. These inequalities may be thought of as martingale models of $Φ$-inequalities for differential operators. The proofs rest on new simple Bellman functions.
We provide sharp bounds for the exponential moments and $p$-moments, $1\leqslant p \leqslant 2$, of the terminate distribution of a martingale whose square function is uniformly bounded by one. We introduce a Bellman function for the corresponding extremal problem and reduce it to the already known Bellman function on $\mathrm{BMO}([0,1])$. In the case of tail estimates, a similar reduction does not work exactly, so we come up with a fine supersolution that leads to sharp tail estimates.
Let $\mathcal{W}$ be a closed dilation and translation invariant subspace of the space of $\mathbb{R}^\ell$-valued Schwartz distributions in $d$ variables. We show that if the space $\mathcal{W}$ does not contain distributions of the type $a\otimes δ_0$, $δ_0$ being the Dirac delta, then the inequality $\|\mathbb{I}_α[f]\|_{L_{p,1}}\lesssim \|f\|_{L_1}$, $\frac{p-1}{p} = \fracα{d}$, holds true for functions $f\in\mathcal{W}\cap L_1$ with a uniform constant; here $\mathbb{I}_α$ is the Riesz potential of order $α$ and $L_{p,1}$ is the Lorentz space. This result implies as a particular case the inequality $\|\nabla^{m-1} f\|_{L_{\frac{d}{d-1},1}} \lesssim \|A f\|_{L_1}$, where $A$ is a canceling elliptic differential operator of order $m$.
We link the problem of estimating the lower Hausdorff dimension of PDE or Fourier constrained measures with Harnack's inequalities for the heat equation. Our approach provides new estimates in the case of Fourier constraints.
We suggest two versions of the Hardy--Littlewood--Sobolev inequality for discrete time martingales. In one version, the fractional integration operator is a martingale transform, however, it may vanish if the filtration is excessively irregular; the second version lacks the martingale property while being analytically meaningful for an arbitrary filtration.
We suggest an elementary Harmonic Analysis approach to canceling and weakly canceling differential operators, which allows to extend these notions to anisotropic setting and also replace differential operators with Fourier multiplies with mild smoothness regularity. In this more general setting of anisotropic Fourier multipliers, we prove the inequality $\|f\|_{L_{\infty}} \lesssim \|Af\|_{L_1}$ if $A$ is a weakly canceling operator of order $d$ and the inequality $\|f\|_{L_2} \lesssim \|Af\|_{L_1}$ if $A$ is a canceling operator of order $\frac{d}{2}$, provided $f$ is a function in $d$ variables.
We provide an estimate from below for the lower Hausdorff dimension of measures on the unit circle based on the arithmetic properties of their spectra. We obtain our bounds via application of a general result for abstract $q$-regular martingales to the Gundy--Varopoulos backwards martingale. To show the sharpness of our method, we improve the best known numerical lower bound for the Hausdorff dimension of certain Riesz products.
We find the best possible constant $C$ in the inequality $\|φ\|_{L^r}\leq C\|φ\|_{L^p}^{\frac{p}{r}}\|φ\|_{\mathrm{BMO}}^{1-\frac{p}{r}}$, where $2 \leq r$ and $p < r$. We employ the Bellman function technique to solve this problem in the case of an interval and then transfer our results to the circle and the line.
We provide martingale analogs of weakly cancelling differential operators and prove a Sobolev-type embedding theorem for these operators in the martingale setting.
We provide a version of the transference principle. It says that certain optimization problems for functions on the circle, the interval, and the line have the same answers. In particular, we show that the sharp constants in the John--Nirenberg inequalities for naturally defined $\mathrm{BMO}$-spaces on the circle, the interval, and the line coincide. The same principle holds true for the Reverse Hölder inequality for Muckenhoupt weights.
We prove a martingale analog of van Schaftingen's theorem and give sharp estimates on the lower Hausdorff dimension of measures in martingale shift invariant spaces. We also provide martingale analogs of trace theorems for Sobolev functions.
We consider several problems related to the restriction of $(\nabla^k) \hat{f}$ to a surface $Σ\subset \mathbb R^d$ with nonvanishing Gauss curvature. While such restrictions clearly exist if $f$ is a Schwartz function, there are few bounds available that enable one to take limits with respect to the $L_p(\mathbb R^d)$ norm of $f$. We establish three scenarios where it is possible to do so: $\bullet$ When the restriction is measured according to a Sobolev space $H^{-s}(Σ)$ of negative index. We determine the complete range of indices $(k, s, p)$ for which such a bound exists. $\bullet$ Among functions where $\hat{f}$ vanishes on $Σ$ to order $k-1$, the restriction of $(\nabla^k) \hat{f}$ defines a bounded operator from (this subspace of) $L_p(\mathbb R^d)$ to $L_2(Σ)$ provided $1 \leq p \leq \frac{2d+2}{d+3+4k}$. $\bullet$ When there is _a priori_ control of $\hat{f}|_Σ$ in a space $H^{\ell}(Σ)$, $\ell > 0$, this implies improved regularity for the restrictions of $(\nabla^k)\hat{f}$. If $\ell$ is large enough then even $\|\nabla \hat{f}\|_{L_2(Σ)}$ can be controlled in terms of $\|\hat{f}\|_{H^\ell(Σ)}$ and $\|f\|_{L_p(\mathbb R^d)}$ alone. The techniques underlying these results are inspired by the spectral synthesis work of Y. Domar, which provides a mechanism for $L_p$ approximation by "convolving along surfaces", and the Stein-Tomas restriction theorem. Our main inequality is a bilinear form bound with similar structure to the Stein--Tomas $T^*T$ operator, generalized to accommodate smoothing along $Σ$ and derivatives transverse to it. It is used both to establish basic $H^{-s}(Σ)$ bounds for derivatives of $\hat{f}$ and to bootstrap from surface regularity of $\hat{f}$ to regularity of its higher derivatives.
We give a simple proof of Dorronsoro's theorem and use similar ideas to establish an equivalence for embeddings of vector fields.
In this paper we develop the method of finding sharp estimates by using a Bellman function. In such a form the method appears in the proofs of the classical John--Nirenberg inequality and $L^p$ estimations of BMO functions. In the present paper we elaborate a method of solving the boundary value problem for the homogeneous Monge--Ampère equation in a parabolic strip for sufficiently smooth boundary conditions. In such a way we have obtained an algorithm of constructing an exact Bellman function for a large class of integral functionals in the BMO space.