arXiv subjects

Francesco Di Plinio

Publications and source records attributed to Francesco Di Plinio.

45 records · Page 3Linked to original sources

arXiv2013

Lacunary Fourier and Walsh-Fourier series near L^1

We prove that, for functions in the Orlicz class LloglogLloglogloglogL, lacunary subsequences of the Fourier and the Walsh-Fourier series converge almost everywhere. Our integrability condition is less stringent than the homologous assumption in the almost everywhere convergence theorems of Lie (Fourier case) and Do-Lacey (Walsh-Fourier case), where the quadruple logarithmic term is replaced by a triple logarithm. Our proof of the Walsh-Fourier case is self-contained and, in antithesis to Do and Lacey's argument, avoids the use of Antonov's lemma, arguing directly via novel weak-L^p bounds for the Walsh-Carleson operator.

math.CA↗

arXiv2013

On weighted norm inequalities for the Carleson and Walsh-Carleson operators

We prove $L^p(w)$ bounds for the Carleson operator ${\mathcal C}$, its lacunary version $\mathcal C_{lac}$, and its analogue for the Walsh series $\W$ in terms of the $A_q$ constants $[w]_{A_q}$ for $1\le q\le p$. In particular, we show that, exactly as for the Hilbert transform, $\|{\mathcal C}\|_{L^p(w)}$ is bounded linearly by $[w]_{A_q}$ for $1\le q<p$. We also obtain $L^p(w)$ bounds in terms of $[w]_{A_p}$, whose sharpness is related to certain conjectures (for instance, of Konyagin \cite{K2}) on pointwise convergence of Fourier series for functions near $L^1$. Our approach works in the general context of maximally modulated Calderón-Zygmund operators.

math.CA↗

arXiv2013

Grisvard's shift theorem near L^infinity and Yudovich theory on polygonal domains

Let Omega be a bounded, simply connected domain with boundary of class C^{1,1} except at finitely many points S_j where the boundary is locally a corner of aperture alpha_j<=pi/2. Improving on results of Grisvard, we show that the solution Gf to the Dirichlet problem on Omega with data f in L^infinity(Omega) and homogeneous boundary conditions has exponentially integrable second derivatives. The proof uses sharp L^p bounds for singular integrals on power weighted spaces inspired by the work of Buckley. Our results allow for the extension of the Yudovich theory of existence, uniqueness and regularity of weak solutions to the Euler equations on Omega x (0,T) to polygonal domains Omega as above.

math.AP↗

arXiv2013

The Euler Equations in planar nonsmooth convex domains

We consider the Euler system set on a bounded convex planar domain, endowed with impermeability boundary conditions. This system is a model for the barotropic mode of the Primitive Equations on a rectangular domain. We show the existence of weak solutions with L^p vorticity for 4/3<= p <= 2, extending and enriching a previous result of Taylor. In the physically interesting case of a rectangular domain, a similar result holds for all 2<p<\infty as well. Moreover, we show the uniqueness of solutions with bounded initial vorticity. The main tool is a new BMO regularity estimate for the Dirichlet problem on domains with corners.

math.AP↗

arXiv2013

The 3-dimensional oscillon equation

On a bounded three-dimensional smooth domain, we consider the generalized oscillon equation with Dirichlet boundary conditions, with time-dependent damping and time-dependent squared speed of propagation. Under structural assumptions on the damping and the speed of propagation, which include the relevant physical case of reheating phase of inflation, we establish the existence of a pullback global attractor of optimal regularity, and finite-dimensionality of the kernel sections.

math.AP↗

arXiv2013

Endpoint bounds for the quartile operator

It is a result by Lacey and Thiele that the bilinear Hilbert transform maps L^{p_1}(R) \times L^{p_2}(R) into L^{p_3}(R) whenever (p_1,p_2,p_3) is a Holder tuple with p_1,p_2 > 1 and p_3>2/3. We study the behavior of the quartile operator, which is the Walsh model for the bilinear Hilbert transform, when p_3=2/3. We show that the quartile operator maps L^{p_1}(R) \times L^{p_2}(R) into L^{2/3,\infty}(R) when p_1,p_2>1 and one component is restricted to subindicator functions. As a corollary, we derive that the quartile operator maps L^{p_1}(R) \times L^{p_2,2/3}(R) into L^{2/3,\infty}(R). We also provide restricted weak-type estimates and boundedness on Orlicz-Lorentz spaces near p_1=1,p_2=2 which improve, in the Walsh case, on results of Bilyk and Grafakos, and Carro-Grafakos-Martell-Soria. Our main tool is the multi-frequency Calderón-Zygmund decomposition first used by Nazarov, Oberlin and Thiele.

math.CA↗

arXiv2012

Logarithmic L^p bounds for maximal directional singular integrals in the plane

We discuss the L^p-boundedness of maximal singular integrals in the plane over a finite set V of N directions. Logarithmic bounds are established for a set V of arbitrary structure in the 2<=p<infinity range. Sharp bounds are proved for lacunary and Vargas sets of directions. The latter include the case of uniformly distributed directions and the finite truncations of the Cantor set. We make use of both classical harmonic analysis methods and product-BMO based time-frequency analysis techniques. As a further application of the latter, we derive an L^p almost orthogonality principle for Fourier restrictions to cones.

math.CA↗

arXiv2010

Time-Dependent Attractor for the Oscillon Equation

We investigate the asymptotic behavior of the nonautonomous evolution problem generated by the Klein-Gordon equation in an expanding background, in one space dimension with periodic boundary conditions, with a nonlinear potential of arbitrary polynomial growth. After constructing a suitable dynamical framework to deal with the explicit time dependence of the energy of the solution, we establish the existence of a regular, time-dependent global attractor. The sections of the attractor at given times have finite fractal dimension.

math-ph↗

arXiv2010

Asymptotics of the Coleman-Gurtin model

This paper is concerned with the integrodifferential equation $$\partial_t u-Δu -\int_0^\infty κ(s)Δu(t-s)\,\d s + φ(u)=f$$ arising in the Coleman-Gurtin's theory of heat conduction with hereditary memory, in presence of a nonlinearity $φ$ of critical growth. Rephrasing the equation within the history space framework, we prove the existence of global and exponential attractors of optimal regularity and finite fractal dimension for the related solution semigroup, acting both on the basic weak-energy space and on a more regular phase space.

math.AP↗