Lp estimates for the wave equation

Let u(x, t) be the solution of u_tt ? Δ_xu = 0 with initial conditions $\text{u(x, 0) = g(x)}\text{and}\text{u}{}_{\mathrm{t}}\text{(x, 0) = ?;(x)}$. Consider the linear operator $\text{T: ?; → u(x, t)}$. (Here g = 0.) We prove for t fixed the following result. Theorem 1: T is bounded in L^p if and only if . Theorem 2: If the coefficients are variables in C and constant outside of some compact set we get: (a) If n = 2k the result holds for $\text{¦ p}{}^{\mathrm{?1}}\text{? 2}{}^{\mathrm{?1}}\text{¦ < (n ? 1)}{}^{\mathrm{?1}}$. (b) If n = 2k ? 1, the result is valid for $\text{¦ p}{}^{\mathrm{?1}}\text{? 2}{}^{\mathrm{?1}}\text{¦ ? (n ? 1)}$. This result are sharp in the sense that for p such that $\text{¦ p}{}^{\mathrm{?1}}\text{? 2}{}^{\mathrm{?1}}\text{¦ > (n ? 1)}{}^{\mathrm{?1}}$ we prove the existence of $\text{?; ? L}{}^{\mathrm{P}}$ in such a way that $\text{T?; ? L}{}^{\mathrm{P}}$. Several applications are given, one of them is to the study of the Klein-Gordon equation, the other to the completion of the study of the family of multipliers $\text{m(ξ) = ψ(ξ) e}{}^{\mathrm{i¦\xi ¦}}\text{¦ ξ ¦}{}^{\mathrm{?b}}$ and finally we get that the convolution against the kernel $\text{K(x) = ?(x)(1 ? ¦ x ¦)}{}^{\mathrm{?1}}$ is bounded in H¹.     

Parabolic maximal functions and potentials of distributions in Hp

A linear homogeneous ODE is constructed, among whose solutions are all products of solutions of two given linear homogeneous ODE's L_m[u]=0, M_n[v]=0, in some classes. Its order is the minimum and its coefficients can be obtained by a finite number of rational operations and differentiations on the coefficients of L_m, M_n. The problem is considered (locally) both in the real and in the complex domain, around an isolated singularity. Examples are also given.     

Spectra of singular measures as multipliers on Lp

Stein's theorem on the interpolation of a family of operators between two analytic spaces is generalized, both to a multiply connected domain and to an interpolation between more than two spaces. The theorem is then applied to get setwise upper bounds for spectra of convolution operators on L^p of the circle. In particular the spectra of operators given by convolution by Cantor-Lebesgue-type measures on L^p are determined. The same is done for certain Riesz products. These results are used to derive a result on translation-invariant subspaces of L^p of the circle.     

One-sided maximal functions and Hp

Let N be the nontangential maximal function of a function u harmonic in the Euclidean half-space Rⁿ × (0, ∞) and let N^? be the nontangential maximal function of its negative part. If u(0, y) = o(y^?n) as y → ∞, then ∥N∥_p ? c_p ∥N^?∥_p, 0 < p < 1, and more. The basic inequality of the paper (Theor. 2.1) can be used not only to derive such global results but also may be used to study the behavior of u near the boundary. Similar results hold for martingales with continuous sample functions. In addition, Theorem 1.3 contains information about the zeros of u. For example, if u belongs to H^p for some 0 < p < 1, then every thick cone in the half-space must contain a zero of u.     

Lp estimates for strongly singular integrals on spaces of homogeneous type

A theory for strongly singular integrals on spaces of homogeneous type is developed and used to obtain sharp multiplier theorems for various groups.     

Large time behavior of the Lp norm of Schrödinger semigroups

LetV ? 0, V?C₀^∞(R^v) with v ? 3 be such that $\text{H = ?}\text{1}\text{2}\text{Δ + V ? 0}$ but for any $\text{ε > 0, ?}\text{1}\text{2}\text{Δ + (1 + ε)V}$ is not positive. We determine the exact rate of divergence of the norm of e^?tH as a map from L^∞ to L^∞. A number of related problems are discussed.     

Trilinear Lp estimates with applications to the Cauchy problem for the Hartree-type equation

In this paper, $L^p$ estimates for a trilinear operator associated with the Hartree type nonlinearity are proved. Moreover, as application of these estimates, it is proved that after a linear transformation, the Cauchy problem for the Hartree-type equation becomes locally well posed in the Bessel potential and homogeneous Besov spaces under certain regularity assumptions on the initial data. This notion of well-posedness and the functional framework to solve the equation were firstly proposed by Y. Zhou.     

Lp Sobolev regularity of averaging operators over hypersurfaces and the Newton polyhedron

$L^p$ to $L_{\beta }^p$ boundedness results are proven for translation invariant averaging operators over hypersurfaces in Euclidean space. The operators can either be Radon transforms or averaging operators with multiparameter fractional integral kernel. In many cases, the amount $\beta >0$ of smoothing proven is optimal up to endpoints, and in such situations this amount of smoothing can be computed explicitly through the use of appropriate Newton polyhedra.     

Proof of the impossibility of the fermat equation Xp + Yp = Zp for special values of p and of the more general equation bXn + cYn = dZn

This paper continues the search to determine for what exponents n Fermat's Last Theorem is true. The main theorem and Corollary 1 consider the set of prime exponents p for which mp + 1 is prime for certain even integers m and prove the truth of FLT in Case 1 for such primes p. The remaining theorems prove the inequality of the more general Fermat equation bXⁿ + cYⁿ = dZⁿ.     

Bi-Sobolev solutions to the prescribed Jacobian inequality in the plane with Lp data and applications to nonlinear elasticity

We construct planar bi-Sobolev mappings whose local volume distortion is bounded from below by a given function $f\in L^p$ with $p>1$. More precisely, for any $1<q<(p+1)/2$ we construct $W^{1,q}$-bi-Sobolev maps with identity boundary conditions; for $f\in L^{\infty }$, we provide bi-Lipschitz maps. The basic building block of our construction are bi-Lipschitz maps which stretch a given compact subset of the unit square by a given factor while preserving the boundary. The construction of these stretching maps relies on a slight strengthening of the celebrated covering result of Alberti, Csörnyei, and Preiss for measurable planar sets in the case of compact sets. We apply our result to a model functional in nonlinear elasticity, the integrand of which features fast blowup as the Jacobian determinant of the deformation becomes small. For such functionals, the derivation of the equilibrium equations for minimizers requires an additional regularization of test functions, which our maps provide.     

The spectra of composition operators on Hp

For H^p, 1 ? p < ∞, composition operators C_?, defined by $\text{C}{}_{\mathrm{?}}\text{(?) = ? ° ?}$ for $\text{? ? H}{}^{\mathrm{p}}$, ? analytic on $\text{D = {z ¦ ¦ z ¦ < 1}}$ are considered, and their spectra determined in the case where ? is analytic on an open region containing D?.     

An upper bound for the zeros of the derivative of Bessel functions

Letj _vk ^′ denotes thekth positive zero of the derivativeJ _v ^′ (x)=dJ _v (x)/dx of Bessel functionJ _v (x) fork=1, 2,…. We establish the upper bound

$$j'_{\nu k}< \nu + a_k \left( {\nu + \frac{{{\rm A}_k^3 }}{{a_k^3 }}} \right)^{\frac{1}{3}} + \frac{3}{{10}}a_k^2 \left( {\nu + \frac{{A_k^3 }}{{a_k^3 }}} \right)^{\frac{1}{3}} , \nu \geqslant 0, k = 1,2, \ldots $$