Related papers: Weighted mixed-norm estimates for circular average…

We prove new $L^p$- $L^q$ estimates for averages over dilates of the circle with respect to $\alpha$-dimensional fractal measure, which unify different types of maximal estimates for the circular average. Our results are consequences of…

This paper is a comprehensive study of $L_p$ estimates for time fractional wave equations of order $\alpha \in (1,2)$ in the whole space, a half space, or a cylindrical domain. We obtain weighted mixed-norm estimates and solvability of the…

We prove an extension of the Stein-Weiss weighted estimates for fractional integrals, in the context of Lp spaces with different integrability properties in the radial and the angular direction. In this way, the classical estimates can be…

We consider Strichartz estimates for the wave equation with respect to general measures which satisfy certain growth condition. In $\mathbb R^{3+1}$ we obtain the sharp estimate and in higher dimensions improve the previous results.

We consider spherical averages of the Fourier transform of fractal measures and improve both the upper and lower bounds on the rate of decay. Maximal estimates with respect to fractal measures are deduced for the Schr\"odinger and wave…

Sharp $L^p$--$L^q$ estimates for the spherical maximal function over dilation sets of fractal dimensions, including the endpoint estimates, were recently proved by Anderson--Hughes--Roos--Seeger. More intricate $L^p$--$L^q$ estimates for…

We consider certain estimates involving averaging operators over curves and hypersurfaces that can be cast into a combinatorial framework. We show that hypersurfaces with nonzero rotational curvature satisfy the usual restricted weak-type…

In this study, utilizing a specific exponential weighting function, we investigate the uniform exponential convergence of weighted Birkhoff averages along decaying waves and delve into several related variants. A key distinction from…

We obtain $L^p-$estimates for the full and lacunary maximal functions associated to the twisted bilinear spherical averages given by \[\mathfrak{A}_t(f_1,f_2)(x,y)=\int_{\mathbb S^{2d-1}}f_1(x+tz_1,y)f_2(x,y+tz_2)\;d\sigma(z_1,z_2),\;t>0,\]…

We prove here essentially sharp linear and bilinear Strichartz type estimates for the wave equations on Minkowski space, where we assume the initial data possesses additional regularity with respect to fractional powers of the usual angular…

Mixed-dimensional elliptic equations exhibiting a hierarchical structure are commonly used to model problems with high aspect ratio inclusions, such as flow in fractured porous media. We derive general abstract estimates based on the theory…

We consider fluctuations of error terms $\Delta(x)$ appearing in the asymptotic formula for a summatory function of coefficients of the Dirichlet series. These are quantified via $\Omega$ and $\Omega_{\pm}$ estimates. We obtain $\Omega$…

The sharp Wolff-type decoupling estimates of Bourgain--Demeter are extended to the variable coefficient setting. These results are applied to obtain new sharp local smoothing estimates for wave equations on compact Riemannian manifolds,…

For any given set $E\subset [1,2]$, we discuss a fractal frequency-localized version of the $L^p$ local smoothing estimates for the half-wave propagator with times in $E$. A conjecture is formulated in terms of a quantity involving the…

We prove (adjoint) bilinear restriction estimates for general phases at different scales in the full non-endpoint mixed norm range, and give bounds with a sharp and explicit dependence on the phases. These estimates have applications to…

Novel global weighted parabolic Sobolev estimates, weighted mixed-norm estimates and a.e. convergence results of singular integrals for evolution equations are obtained. Our results include the classical heat equation, the harmonic…

We establish near-optimal mixed-norm estimates for the X-ray transform restricted to polynomial curves with a weight that is a power of the affine arclength. The bounds that we establish depend only on the spatial dimension and the degree…

We study maximal estimates for the wave equation with orthonormal initial data. In dimension $d=3$, we establish optimal results with the sharp regularity exponent up to the endpoint. In higher dimensions $d \ge 4$ and also in $d=2$, we…

We consider the averages of a function $ f$ on $ \mathbb R ^{n}$ over spheres of radius $ 0< r< \infty $ given by $ A_{r} f (x) = \int_{\mathbb S ^{n-1}} f (x-r y) \; d \sigma (y)$, where $ \sigma $ is the normalized rotation invariant…

We prove variation-norm estimates for certain oscillatory integrals related to Carleson's theorem. Bounds for the corresponding maximal operators were first proven by Stein and Wainger. Our estimates are sharp in the range of exponents, up…