Martingale Hardy spaces and the dyadic derivative

It is shown that the maximal operator of the one-dimensional dyadic derivative of the dyadic integral is bounded from the dyadic Hardy-Lorentz spaceH _p,q toL _p,q (1/2<p<∞, 0<q≤∞) and is of weak type (L ₁,L ₁). We define the twodimensional dyadic hybrid Hardy spaceH ₁ ^‖ and verify that the corresponding maximal operator of a two-dimensional function is of weak type (H ₁ ^‖ ,L ₁). As a consequence, we obtain that the dyadic integral of a two-dimensional functionfεH ₁ ^‖ ?LlogL is dyadically differentiable and its derivative is a.e.f.     

Growth rate of the functions in Bergman type spaces

For 0<p,α<∞, let ‖f‖_p,α be the L^p-norm with respect the weighted measure . We define the weighted Bergman space A_α^p(D) consisting of holomorphic functions f with ‖f‖_p,α<∞. For any σ>0, let A^−σ(D) be the space consisting of holomorphic functions f in D with . If D has C² boundary, then we have the embedding A_α^p(D)⊂A^−(n+α)/p(D). We show that the condition of C²-smoothness of the boundary of D is necessary by giving a counter-example of a convex domain with C^1,λ-smooth boundary for 0<λ<1 which does not satisfy the embedding.     

On the Robin Boundary Condition for Laplace's Equation in Lipschitz Domains

Abstract

Let Ω be a bounded Lipschitz domain in ? ⁿ , n ≥ 3 with connected boundary. We study the Robin boundary condition ?u/?N + bu = f ∈ L ^p (?Ω) on ?Ω for Laplace's equation Δu = 0 in Ω, where b is a non-negative function on ?Ω. For 1 < p < 2 + ?, under suitable compatibility conditions on b, we obtain existence and uniqueness results with non-tangential maximal function estimate ‖(?u)*‖ _p  ≤ C‖f‖ _p , as well as a pointwise estimate for the associated Robin function. Moreover, the solution u is represented by a single layer potential.     

Upper and lower bounds on norms of functions of matrices

Given an n by n matrix A, we look for a set S in the complex plane and positive scalars m and M such that for all functions p bounded and analytic on S and throughout a neighborhood of each eigenvalue of A, the inequalities

m·inf{‖f‖_L^∞(S):f(A)=p(A)}?‖p(A)‖?M·inf{‖f‖_L^∞(S):f(A)=p(A)}     

Weighted tight frames of exponentials on a finite interval

Given a finite intervalI?R, a characterization is given for those discrete sets of real numbers Λ and associated sequences {c _λ}_λ∈Λ, withc _λ>0, having the properties that every functionf∈L ²(I) can be expanded inL ²(I) as the unconditionally convergent series $$f = \sum\limits_{\lambda \in \Lambda } {\hat f} (\lambda )c_\lambda e^{2\pi i\lambda x} $$ and that the range of the mappingL ²(I)→L _μ ² :f→f has finite codimension inL _μ ² , iff denotes the Fourier transform off and μ is the measure μ = ∑_λ∈Λ c _λ δ_λ.     

Blowup stability of solutions of the nonlinear heat equation with a large life span

We study the Cauchy problem for the nonlinear heat equation u_t-?u=|u|^p-1u in R^N. The initial data is of the form u₀=λ?, where ?∈C₀(R^N) is fixed and λ>0. We first take 1<p<p_f, where p_f is the Fujita critical exponent, and ?∈C₀(R^N)∩L¹(R^N) with nonzero mean. We show that u(t) blows up for λ small, extending the H. Fujita blowup result for sign-changing solutions. Next, we consider 1<p<p_s, where p_s is the Sobolev critical exponent, and ?(x) decaying as |x|^-σ at infinity, where p<1+2/σ. We also prove that u(t) blows up when λ is small, extending a result of T. Lee and W. Ni. For both cases, the solution enjoys some stable blowup properties. For example, there is single point blowup even if ? is not radial.     

A grothendieck factorization theorem on 2-convex schatten spaces

We prove that for every bounded linear operatorT:C ^2p →H(1≤p<∞,H is a Hilbert space,C ^{2 p} p is the Schatten space) there exists a continuous linear formf onC ^p such thatf≥0, ‖f‖(C _C ^p)*=1 and $$\forall x \in C^{2p} , \left\| {T(x)} \right\| \leqslant 2\sqrt 2 \left\| T \right\|< f\frac{{x * x + xx*}}{2} > 1/2$$ . Forp=∞ this non-commutative analogue of Grothendieck’s theorem was first proved by G. Pisier. In the above statement the Schatten spaceC ^2p can be replaced byE _E ² whereE ⁽²⁾ is the 2-convexification of the symmetric sequence spaceE, andf is a continuous linear form onC _E. The statement can also be extended toL _E{(su2)}(M, τ) whereM is a Von Neumann algebra,τ a trace onM, E a symmetric function space.     

Sharp bounds for integral means

We introduce a bound M of f, ‖f‖_∞?M?2‖f‖_∞, which allows us to give for 0?p<∞ sharp upper bounds, and for −∞<p<0 sharp lower bounds for the average of |f|^p over E if the average of f over E is zero. As an application we give a new proof of Grüss's inequality estimating the covariance of two random variables. We also give a new estimate for the error term in the trapezoidal rule.     

Unique continuation for Schrodinger operators with unbounded potentials

We consider unique continuation theorems for solution of inequalities $\text{¦Δu(x)¦ ? W(x) ¦u(x)¦}$ with W allowed to be unbounded. We obtain two kinds of results. One allows W ? L^p_loc($\text{R}$ⁿ) with $\text{p ? n ? 2}\text{for}\text{n > 5, p >}\text{1}\text{3}\text{(2n ? 1)}\text{for}\text{n ? 5}$. The other requires fW² to be ?Δ-form bounded for all f ? C₀^∞.     

The Stokes-system in exterior domains:L p -estimates for small values of a resolvent parameter

We consider the resolvent problem for the Stokes-system in an exterior domain: $$- v \cdot \Delta u + \lambda \cdot u + \nabla \pi = f,divu = 0in\mathbb{R}^3 \backslash \bar \Omega ,$$ , with υε]0, ∞[, λε?] ?∞, 0], Ω bounded domain in ?³, withC ²-boundary ?Ω. In addition, Dirichlet boundary conditionsu¦?Ω=0 are prescribed. Using the method of integral equations, we estimate solutions (u,π) inL ^p -norms, for small values of ¦λ¦.