Following the ideas of Andrei Lerner in [ A pointwise estimate for the local sharp maximal function with applications to singular integrals" Bull. London Math. Soc. 42 (2010) 843856], we obtain another decomposition of an arbitrary measurable function f in terms of local mean oscillations. This allows us to get new estimates involving one-sided singular integrals and one-sided maximal operator. As an application to this result we obtain two weighted inequality for one-sided singular integrals and a L(w) inequality relating a measurable function f and sharp one-sided operator. These estimates are more precise in sense that they are valid for a greater class of weights.
On a smooth bounded domain Ω\subset R^N we consider the Schrödinger operators -Δ-V, with V being either the critical borderline potential V(x)=(N-2)^2/4 |x|^{-2} or V(x)=(1/4) dist (x,\partialΩ)^{-2}, under Dirichlet boundary conditions. In this work we obtain sharp two-sided estimates on the corresponding heat kernels. To this end we transform the Scrödinger operators into suitable degenerate operators, for which we prove a new parabolic Harnack inequality up to the boundary. To derive the Harnack inequality we have established a serier of new inequalities such as improved Hardy, logarithmic …
We consider non degenerate Brownian SDEs with H{ö}lder continuous in space diffusion coefficient and unbounded drift with linear growth. We derive two sided bounds for the associated density and pointwise controls of its derivatives up to order two under some additional spatial H{ö}lder continuity assumptions on the drift. Importantly, the estimates reflect the transport of the initial condition by the unbounded drift through an auxiliary, possibly regularized, flow.
We prove existence of the global attractor of the damped and driven 2D Euler--Bardina equations on the torus and give an explicit two-sided estimate of its dimension that is sharp as $α\to0^+$.
We improve the recent result of Chae & Tadmor in [Comm. Math. Sci. 6(3) (2008) 785-789], proving a one-sided threshold condition which leads to finite-time breakdown of the Euler-Poisson equations in arbitrary dimension n.
We prove on-diagonal bounds for the heat kernel of the Dirichlet Laplacian $-Δ^D_Ω$ in locally twisted three-dimensional tubes $Ω$. In particular, we show that for any fixed $x$ the heat kernel decays for large times as $\mathrm{e}^{-E_1t}\, t^{-3/2}$, where $E_1$ is the fundamental eigenvalue of the Dirichlet Laplacian on the cross section of the tube. This shows that any, suitably regular, local twisting speeds up the decay of the heat kernel with respect to the case of straight (untwisted) tubes. Moreover, the above large time decay is valid for a wide class of subcritical operators defined…
We establish sharp two-sided weighted bounds on the fundamental solution to the fractional Schrödinger operator using the method of desingularizing weights.
We prove a sharp Hölder estimate for solutions of linear two-dimensional, divergence form elliptic equations with measurable coefficients, such that the matrix of the coefficients is symmetric and has {\em unit determinant}. Our result extends some previous work by Piccinini and Spagnolo. The proof relies on a sharp Wirtinger type inequality.
We provide sharp two-sided estimates on the Dirichlet heat kernel $k_1(t,x,y)$ for the Laplacian in a ball. The result accurately describes the exponential behaviour of the kernel for small times and significantly improves the qualitatively sharp results known so far. As a consequence we obtain the full description of the kernel $k_1(t,x,y)$ in terms of its global two-sided sharp estimates.
In this paper, we establish sharp two-sided estimates for transition densities of a large class of subordinate Markov processes. As applications, we show that the parabolic Harnack inequality and Hölder regularity hold for parabolic functions of such processes, and derive sharp two-sided Green function estimates.
The Oleinik inequality for conservation laws and Aronson-Benilan type inequalities for porous medium or p-Laplacian equations are one-sided inequalities that provide the fundamental features of the solution such as the uniqueness and sharp regularity. In this paper such one-sided inequalities are unified and generalized for a wide class of first and second order equations in the form of $$ u_t=σ(t,u,u_x,u_{xx}),\quad u(x,0)=u^0(x)\ge0,\quad t>0,\,x\in\bfR, $$ where the non-strict parabolicity ${\partial\over\partial q} σ(t,z,p,q)\ge0$ is assumed. The generalization or unification of one-sided …
In the present paper we shall establish n-dimensional Hardy's inequalities with non-doubling weight functions of the distance to the boundary, where the boundary is a $C^2$ class bounded domain of $R^N$. This work is essentially based on one dimensional weighted Hardy's inequalities with one-sided boundary condition and sharp remainders. As weights we admit rather general ones that may vanish or blow up in infinite order such as $e^{-1/t}$ or $e^{1/t}$ at $t=0$ in one dimensional case.
By using Duhamel's formula, we prove sharp two-sided estimates for the heat kernel of spectral fractional Laplacian with time-dependent gradient perturbation in bounded $C^{1,1}$ domains. Moreover, we also obtain gradient estimate as well as Hölder continuity of the gradient of the heat kernel.
We consider the follow-the-leader particle approximation scheme for a $1d$ scalar conservation law with nonnegative $L^\infty_c$ initial datum and with a $C^1$ concave flux, which is known to provide convergence towards the entropy solution $ρ$ to the corresponding Cauchy problem. We provide two novel contributions to this theory. First, we prove that the one-sided Lipschitz condition satisfied by the approximating density $ρ^n$ is a discrete version of an entropy condition; more precisely, under fairly general assumptions on $f$ (which imply concavity of $f$) we prove that the continuum versi…
In this work we construct the heat kernel of the 1/2-order Laplacian perturbed by the first-order gradient term in Hölder space and the zero-order potential term in generalized Kato's class, and obtain sharp two-sided estimates as well as the gradient estimate of the heat kernel.
In this note we give a sharp weighted estimate for square function from $L^2(w)$ to $L^2(w)$, $w\in A_2$. This has been known. But we also give a sharpening of this weighted estimate in the spirit of $T1$-type testing conditions. Finally we show that for any weight $w\in A^d_2$ and any characteristic function of a measurable set $\|S_wχ_E\|_{L^{2, \infty}(w^{-1})} \le C \sqrt{[w]_{A^d_2}}\, \|χ_E\|_w$, and this estimate is sharp. So on characteristic functions of measurable sets at least, no logarithmic correction is needed for the weak type of the dyadic square function.The sharp estimate for…
We extend Krylov and Röckner's result \cite{KR} to the drift coefficients in critical Lebesgue space, and prove the existence and uniqueness of weak solutions for a class of SDEs. To be more precise, let $b: [0,T]\times{\mathbb R}^d\rightarrow{\mathbb R}^d$ be Borel measurable, where $T>0$ is arbitrarily fixed. Consider $$X_t=x+\int_0^tb(s,X_s)ds+W_t,\quad t\in[0,T], \, x\in{\mathbb R}^d,$$ where $\{W_t\}_{t\in[0,T]}$ is a $d$-dimensional standard Wiener process. If $b=b_1+b_2$ such that $b_1(T-\cdot)\in\mathcal{C}_q^0((0,T];L^p({\mathbb R}^d))$ with $2/q+d/p=1$ for $p,q\ge1$ and $\|b_1(T-\cdo…
We consider a nonvariational degenerate elliptic operator structured on a system of left invariant, 1-homogeneous, Hörmander's vector fields on a Carnot group in $R^{n}$, where the matrix of coefficients is symmetric, uniformly positive on a bounded domain of $R^{n}$ and the coefficients are bounded, measurable and locally VMO in the domain. We give a new proof of the interior $L^{p}$ estimates on the second order derivatives with respect to the vector fields, first proved by Bramanti-Brandolini in [Rend. Sem. Mat. dell'Univ. e del Politec. di Torino, Vol. 58, 4 (2000), 389-433], extending to …
Consider the Schrödinger operator $ \mathcal L^V=-Δ+V $ on $\R^d$, where $V:\R^d\to [0,\infty)$ is a nonnegative and locally bounded potential on $\R^d$ so that for all $x\in \R^d$ with $|x|\ge 1$, $c_1g(|x|)\le V(x)\le c_2g(|x|)$ with some constants $c_1,c_2>0$ and a nondecreasing and strictly positive function $g:[0,\infty)\to [1,+\infty)$ that satisfies $g(2r)\le c_0 g(r)$ for all $r>0$ and $\lim_{r\to \infty} g(r)=\infty.$ We establish global in time and qualitatively sharp bounds for the heat kernel of the associated Schrödinger semigroup by the probabilistic method. In particular, we can…
In this paper we obtain for $T^+$, a one-sided singular integral given by a Calderón-Zygmund kernel with support in $(-\infty,0)$, a $L^p(w)$ bound when $w\in A_1^+$. A. K. Lerner, S. Ombrosi, and C. Pérez proved in [ "$A_{1}$ Bounds for Calderón-Zygmund operators related to a problem of Muckenhoupt and Wheeden", Math. Res. Lett. \textbf{16} (2009), no. 1, 149-156] that this bound is sharp with respect to $||w||_{A_1} $ and $p$ . We also give a $L^{1,\infty}(w)$ estimate, for a related problem of Muckenhoupt and Wheeden for $w\in A_1^+$ . We improve the classical results, for one-sided singula…
Consider the heat equation with a nonlinear boundary condition $$ \partial_t u=Δu,\quad x\in{\bf R}^N_+,\,\,\,t>0,\qquad \partial_νu=u^p, \quad x\in\partial{\bf R}^N_+,\,\,\,t>0,\qquad u(x,0)=κψ(x),\quad x\in D:=\overline{{\bf R}^N_+}, $$ where $N\ge 1$, $p>1$, $κ>0$ and $ψ$ is a nonnegative measurable function in ${\bf R}^N_+ :=\{y\in{\bf R}^N:y_N>0 \}$. Let us denote by $T(κψ)$ the life span of solutions to this problem. We investigate the relationship between the singularity of $ψ$ at the origin and $T(κψ)$ for sufficiently large $κ>0$ and the relationship between the behavior of $ψ$ at the…
We establish sharp two-sided bounds on the heat kernel of the fractional Laplacian, perturbed by a drift having critical-order singularity, by transferring it to appropriate weighted space with singular weight.
We obtain a local estimate for the gradient of solutions to a second-order elliptic equation in divergence form with bounded measurable coefficients that are square-Dini continuous at the single point x=0. In particular, we treat the case of solutions that are not Lipschitz continuous at x=0. We show that our estimate is sharp.
We give again (see also arXiv:1112.0676) a proof of weighted estimate of any Calderón-Zygmund operator. This is under a universal sharp sufficient condition that is weaker than the so-called bump condition. Bump conjecture was recently solved independently and simultaneously by A. Lerner (arXiv:1202.1860) and Nazarov--Reznikov-Treil-Volberg (arXiv:1202.2406). The latter paper uses the Bellman approach. Immediately a very natural and seemingly simple question arises how to strengthen the bump conjecture result by weakening its assumptions in a natural symmetric way. This is what we are dealing …
We provide a sharp double-sided estimate for Poincaré-Sobolev constants on a convex set, in terms of its inradius and $N-$dimensional measure. Our results extend and unify previous works by Hersch and Protter (for the first eigenvalue) and of Makai, Pólya and Szegő (for the torsional rigidity), by means of a single proof.
In the first part of this article, we prove two-sided estimates of hitting probabilities of balls, the potential kernel and the Green function for a ball for general isotropic unimodal Lévy processes. Our bounds are sharp under the absence of the Gaussian component and a mild regularity condition on the density of the Lévy measure: its radial profile needs to satisfy a scaling-type condition, which is equivalent to $O$-regular variation at zero and at infinity with lower indices greater than $-d - 2$. We also prove a supremum estimate and a regularity result for functions harmonic with respect…
We give sharp two-sided estimates of the semigroup generated by the fractional Laplacian plus the Hardy potential on $\mathbb{R}^d$, including the case of the critical constant. We use Davies' method back-to-back with a new method of integral analysis of Duhamel's formula.
We establish sharp $L^2$-Sobolev estimates for classes of pseudodifferential operators with singular symbols whose non-pseudodifferential (Fourier integral operator) parts exhibit two-sided fold singularities. The operators considered include both singular integral operators along curves in $R^2$ with simple inflection points and normal operators arising in linearized seismic imaging in the presence of fold caustics.
