arXiv subjects

Philip T. Gressman

Publications and source records attributed to Philip T. Gressman.

31 records · Page 2Linked to original sources

arXiv2012

Fractional Poincare and logarithmic Sobolev inequalities for measure spaces

We prove generalizations of the Poincare and logarithmic Sobolev inequalities corresponding to the case of fractional derivatives in measure spaces with only a minimal amount of geometric structure. The class of such spaces includes (but is not limited to) spaces of homogeneous type with doubling measures. Several examples and applications are given, including Poincare inequalities for graph Laplacians, Fractional Poincare inequalities of Mouhot, Russ, and Sire [16], and implications for recent work of the author and R. M. Strain on the Boltzmann collision operator [10, 11, 9].

math.CA↗

arXiv2012

Scalar oscillatory integrals in smooth spaces of homogeneous type

We consider a generalization of the notion of spaces of homogeneous type, inspired by recent work of Street [21] on the multi-parameter Carnot-Caratheodory geometry, which imbues such spaces with differentiability structure. The setting allows one to formulate estimates for scalar oscillatory integrals on these spaces which are uniform and respect the underlying geometry of both the space and the phase function. As a corollary we obtain a generalization of a theorem of Bruna, Nagel, and Wainger [1] on the asymptotic behavior of scalar oscillatory integrals with smooth, convex phase of finite type.

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arXiv2012

A non-local inequality and global existence

In this article we prove a collection of new non-linear and non-local integral inequalities. As an example for $u\ge 0$ and $p\in (0,\infty)$ we obtain $$ \int_{\threed} dx ~ u^{p+1}(x) \le (\frac{p+1}{p})^2 \int_{\threed} dx ~ \{(-\triangle)^{-1} u(x) \} \nsm \nabla u^{\frac{p}{2}}(x)\nsm^2. $$ We use these inequalities to deduce global existence of solutions to a non-local heat equation with a quadratic non-linearity for large radial monotonic positive initial conditions. Specifically, we improve \cite{ksLM} to include all $α\in (0, 74/75)$.

math.AP↗

arXiv2011

Sharp anisotropic estimates for the Boltzmann collision operator and its entropy production

This article provides sharp constructive upper and lower bound estimates for the non-linear Boltzmann collision operator with the full range of physical non cut-off collision kernels ($γ> -n$ and $s\in (0,1)$) in the trilinear $L^2(\R^n)$ energy $<\mathcal{Q}(g,f),f>$. These new estimates prove that, for a very general class of $g(v)$, the global diffusive behavior (on $f$) in the energy space is that of the geometric fractional derivative semi-norm identified in the linearized context in our earlier works [2009, 2010, 2010 arXiv:1011.5441v1]. We further prove new global entropy production estimates with the same anisotropic semi-norm. This resolves the longstanding, widespread heuristic conjecture about the sharp diffusive nature of the non cut-off Boltzmann collision operator in the energy space $L^2(\R^n)$.

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arXiv2010

Global Classical Solutions of the Boltzmann Equation with Long-Range Interactions and Soft Potentials

In this work we prove global stability for the Boltzmann equation (1872) with the physical collision kernels derived by Maxwell in 1866 for the full range of inverse power intermolecular potentials, $r^{-(p-1)}$ with $p>2$. This completes the work which we began in (arXiv:0912.0888v1). We more generally cover collision kernels with parameters $s\in (0,1)$ and $γ$ satisfying $γ> -(n-2)-2s$ in arbitrary dimensions $\mathbb{T}^n \times \mathbb{R}^n$ with $n\ge 2$. Moreover, we prove rapid convergence as predicted by the Boltzmann H-Theorem. When $γ+ 2s \ge 0$, we have exponential time decay to the Maxwellian equilibrium states. When $γ+ 2s < 0$, our solutions decay polynomially fast in time with any rate. These results are constructive. Additionally, we prove sharp constructive upper and lower bounds for the linearized collision operator in terms of a geometric fractional Sobolev norm; we thus observe that a spectral gap exists only when $γ+ 2s \ge 0$, as conjectured in Mouhot-Strain (2007).

math.AP↗

arXiv2010

Global Strong Solutions of the Boltzmann Equation without Angular Cut-off

We prove the existence and exponential decay of global in time strong solutions to the Boltzmann equation without any angular cut-off, i.e., for long-range interactions. We consider perturbations of the Maxwellian equilibrium states and include the physical cross-sections arising from an inverse-power intermolecular potential $r^{-(p-1)}$ with $p>3$, and more generally, the full range of angular singularities $s=ν/2 \in(0,1)$. These appear to be the first unique global solutions to this fundamentally important model, which grants a basic example where a range of geometric fractional derivatives occur in a physical model of the natural world. Our methods provide a new understanding of the effects of grazing collisions in the Boltzmann theory.

math.AP↗

arXiv2010

Uniform sublevel Radon-like inequalities

This paper is concerned with establishing uniform weighted $L^p$-$L^q$ estimates for a class of operators generalizing both Radon-like operators and sublevel set operators. Such estimates are shown to hold under general circumstances whenever a scalar inequality holds for certain associated measures (the inequality is of the sort studied by Oberlin, relating measures of parallelepipeds to powers of their Euclidean volumes). These ideas lead to previously unknown, weighted affine-invariant estimates for Radon-like operators as well as new $L^p$-improving estimates for degenerate Radon-like operators with folding canonical relations which satisfy an additional curvature condition of Greenleaf and Seeger for FIOs (building on the ideas of Sogge and Mockenhaupt, Seeger, and Sogge); these new estimates fall outside the range of estimates which are known to hold in the generality of the FIO context.

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arXiv2009

On multilinear determinant functionals

This paper considers the problem of $L^p$-estimates for a certain multilinear functional involving integration against a kernel with the structure of a determinant. Examples of such objects are ubiquitous in the study of Fourier restriction and geometric averaging operators. It is shown that, under very general circumstances, the boundedness of such functionals is equivalent to a geometric inequality for measures which has recently appeared in work by D. Oberlin (Math Proc. Cambridge. Philos. Soc., 129, 2000) and Bak, Oberlin, and Seeger (J. Aust. Math. Soc., 85, 2008).

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arXiv2009

Uniform geometric estimates for sublevel sets

This paper reconsiders the uniform sublevel set estimates of Carbery, Christ, and Wright (1999) and Phong, Stein, and Sturm (2001) from a geometric perspective. This perspective leads one to consider a natural collection of homogeneous, nonlinear differential operators which generalize mixed derivatives in $\R^d$. As a consequence, it is shown that, in the case of both of these previous works, improved uniform decay rates are possible in many situations.

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arXiv2008

$L^p$-improving estimates for averages on polynomial curves

In the combinatorial method proving of $L^p$-improving estimates for averages along curves pioneered by Christ (IMRN, 1998), it is desirable to estimate the average modulus (with respect to some uniform measure on a set) of a polynomial-like function from below using only the value of the function or its derivatives at some prescribed point. In this paper, it is shown that there is always a relatively large set of points (independent of the particular function to be integrated) for which such estimates are possible. Inequalities of this type are then applied to extend the results of Tao and Wright (JAMS, 2003) to obtain endpoint restricted weak-type estimates for averages over curves given by polynomials.

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arXiv2008

Rank and regularity for averages over submanifolds

This paper establishes endpoint $L^p-L^q$ and Sobolev mapping properties of Radon-like operators which satisfy a homogeneity condition (similar to semiquasihomogeneity) and a condition on the rank of a matrix related to rotational curvature. For highly degenerate operators, the rank condition is generically satisfied for algebraic reasons, similar to an observation of Greenleaf, Pramanik, and Tang concerning oscillatory integral operators.

math.CA↗

arXiv2007

Uniform estimates for cubic oscillatory integrals

This paper establishes the optimal decay rate for scalar oscillatory integrals in $n$ variables which satisfy a nondegeneracy condition on the third derivatives. The estimates proved are stable under small linear perturbations, as encountered when computing the Fourier transform of surface-carried measures. The main idea of the proof is to construct a nonisotropic family of balls which locally capture the scales and directions in which cancellation occurs.

math.CA↗

arXiv2007

Sharp $L^p$-$L^q$ estimates for generalized $k$-plane transforms

In this paper, optimal $L^p-L^q$ estimates are obtained for operators which average functions over polynomial submanifolds, generalizing the $k$-plane transform. An important advance over previous work is that full $L^p-L^q$ estimates are obtained by methods which have traditionally yielded only restricted weak-type estimates. In the process, one is lead to make coercivity estimates for certain functionals on $L^p$ for $p < 1$.

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