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.     

Hardy spaces and the<Emphasis Type="Italic">Tb</Emphasis> theorem

It is well-known that Calderón-Zygmund operators T are bounded on H^p for\(\frac{n}{{n + 1}}< p \leqslant 1\) provided T*(1) = 0. In this article, it is shown that if T*(b) = 0, where b is a para-accretive function, T is bounded from the classical Hardy space H^p to a new Hardy space H _b ^p . To develop an H _b ^p theory, a discrete Calderón-type reproducing formula and Plancherel-Pôlya-type inequalities associated to a para-accretive function are established. Moreover, David, Journé, and Semmes’ result [9] about the L^P, 1 < p < ∞, boundedness of the Littlewood-Paley g function associated to a para-accretive function is generalized to the case of p ≤ 1. A new characterization of the classical Hardy spaces by using more general cancellation adapted to para-accretive functions is also given. These results complement the celebrated Calderón-Zygmund operator theory.     

Maximal operators, Riesz transforms and Littlewood-Paley functions associated with Bessel operators on BMO

In this paper we study boundedness properties of certain harmonic analysis operators (maximal operators for heat and Poisson semigroups, Riesz transforms and Littlewood-Paley g-functions) associated with Bessel operators, on the space BMOo(R) that consists of the odd functions with bounded mean oscillation on R.     

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.     

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?.     

A Fatou theorem for α-harmonic functions

We study functions which are harmonic in the upper half space with respect to (−Δ)^α/2, 0<α<2. We prove a Fatou theorem when the boundary function is L^p-Hölder continuous of order β and βp>1. We give examples to show this condition is sharp.     

Littlewood-Paley and Lusin functions on nilpotent Lie groups

In this paper, we define the Littlewood-Paley and Lusin functions associated to the sub-Laplacian operator on nilpotent Lie groups. Then we prove the Lp (1<p<∞) boundedness of Littlewood-Paley and Lusin functions.     

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.     

Lusin type and cotype for Laguerre <Emphasis Type="Italic">g</Emphasis>-functions

We characterize Lusin type and cotype for a Banach space in terms of the L ^p -boundedness of Littlewood-Paley g-functions associated with the Hermite and Laguerre expansions.     

On l p -multipliers of functions analytic in the disk

We consider bounded analytic functions in domains generated by sets with the Littlewood-Paley property. We show that each such function is an l ^p -multiplier.     

On equivalence of L p -norms related to Schrödinger type operators on Riemannian manifolds

In this paper, we study Schrödinger type operator on a Riemannian manifold. Under some assumptions on a potential function, we characterize the domain of the square root of the Schrödinger type operator on L ^p space. In the proof, the defective intertwining properties and the Littlewood-Paley inequalities play important roles.     

On some special codes over Fp + uFp + uFp

We define a new map between codes over F_p + uF_p + u²F_p and F_p which is different to that defined in [2]. It is proved that the image of the linear cyclic code over the commutative ring F_p + uF_p + u²F_p with length n under this map is a distance-invariant quasi-cyclic code of index p² with length p²n over F_p. Moreover, it is proved that, if (n, p) = 1, then every code with length p²n over F_p which is the image of a linear (1 − u²)-cyclic code with length n over F_p + uF_p + u²F_p under this map is permutation equivalent to a quasi-cyclic code of index p².     

Quantum complexity of the approximation for the classes B(Wp ([0, 1])) and B(Hp ([0, 1]))

We study the approximation of functions from anisotropic Sobolev classes B(W^r_p([0,1]d)) and Hölder-Nikolskii classes B(H^r_p([0,1]d)) in the L_q([0,1]d) norm with q ≤ p in the quantum model of computation. We determine the quantum query complexity of this problem up to logarithmic factors. It shows that the quantum algorithms are significantly better than the classical deterministic or randomized algorithms.     

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.     

Conjugate function theory in weak1 Dirichlet algebras

The fundamental theorems on conjugate functions are shown to be valid for weak¹ Dirichlet algebras. In particular the conjugation operator is shown to be a continuous map of L^p to L^p for 1 < p < ∞, to be a continuous map of L¹ to L^p, 0 < p < 1, and to map functions in L^∞ to exponentially integrable functions. These results allow a number of results for Dirichlet algebras to be extended to weak¹ Dirichlet algebras.     

Some weighted estimates for Littlewood-Paley functions and radial multipliers

We prove some weighted estimates for certain Littlewood-Paley operators on the weighted Hardy spaces H_w^p (0<p?1) and on the weighted L^p spaces. We also prove some weighted estimates for the Bochner-Riesz operators and the spherical means.     

Distance to Ck hypersurfaces

It is shown by elementary means that a C^k hypersurface M of positive reach in $\text{R}$^{n + 1} has the property that the signed distance function to it is C^k, k ? 1. This extends and complements work of Federer, Gilbarg and Trudinger, and Serrin.     

Wr,p(R)-splines

In [3] Golomb describes, for 1 < p < ∞, the H^r,p(R)-extremal extension $\text{F}{}_1$ of a function $\text{?:E → R}$ (i.e., the H^r,p-spline with knots in E) and studies the cone $\text{H}{}_{1E}{}^{\mathrm{r,p}}$ of all such splines. We study the problem of determining when $\text{F}{}_1$ is in W^r,p ≡ H^r,p ∩ L^p. If $\text{F}{}_1\text{? W}{}^{\mathrm{r,p}}$, then $\text{F}{}_1$ is called a W^r,p-spline, and we denote by $\text{W}{}_{1E}{}^{\mathrm{r,p}}$ the cone of all such splines. If E is quasiuniform, then $\text{F}{}_1\text{? W}{}^{\mathrm{r,p}}$ if and only if . The cone $\text{W}{}_{1E}{}^{\mathrm{r,p}}$ with E quasiuniform is shown to be homeomorphic to l^p. Similarly, $\text{H}{}_{1E}{}^{\mathrm{r,p}}$ is homeomorphic to h^r,p. Approximation properties of the W^r,p-splines are studied and error bounds in terms of the mesh size $\text{¦ E ¦}$ are calculated. Restricting ourselves to the case p = 2 and to quasiuniform partitions E, the second integral relation is proved and better error bounds in terms of $\text{¦ E ¦}$ are derived.     

Inequalities related toH p smoothness of Sobolev type

By exploiting a class of maximal functions and Littlewood-Paley theory, a list of embedding inequalities onH ^p-Sobolev spaces andH ^p boundedness results for Riesz and Bessel potentials are obtained at one stroke.This work was supported in part by the Chung-Ang University Academic Research Special Grants, 1997.     

Subgroups of LF(2, 2n)

J. Gierster [Math. Ann. 26, 309–368] has proven a number of theorems giving the subgroup structure of LF(2, pⁿ), p an odd prime, and has given simple necessary and sufficient conditions for two elements of LF(2, pⁿ) to be conjugate. In this paper we obtain analogous theorems and conditions for LF(2, 2ⁿ). The methods involve solving congruences mod 2ⁿ.