A COUNTEREXAMPLE INLp APPROXIMATION BY HARMONIC FUNCTIONS (original) (raw)
𝐿𝑖𝑝𝛼 Harmonic Approximation on Closed Sets
Proceedings of the American Mathematical Society, 2001
In this paper the L i p α Lip\alpha harmonic approximation ( 0 > α > 1 2 0 > \alpha > \frac {1}{2} ) on relatively closed subsets of a domain in the complex plane is characterized under the same conditions given by S. Gardiner for the uniform case. Thus, the result of P. Paramonov on L i p α Lip\alpha harmonic polynomial approximation for compact subsets is extended to closed sets. Moreover, the problem of uniform approximation with continuous extension to the boundary for harmonic functions and similar questions in L i p α Lip\alpha harmonic approximation are also studied.
Harmonic type approximation lemmas
Journal of Mathematical Analysis and Applications, 2009
We give a survey of known and not known harmonic type approximation lemmas which are descendants of the classical De Giorgi's one, and we outline some of their recent or possible applications.
Local compactness for families of {$\scr A$}-harmonic functions
Illinois Journal of Mathematics, 2004
We show that if a family of A-harmonic functions that admits a common growth condition is closed in L p loc , then this family is locally compact on a dense open set under a family of topologies, all generated by norms. This implies that when this family of functions is a vector space, then such a vector space of A-harmonic functions is finite dimensional if and only if it is closed in L p loc. We then apply our theorem to the family of all p-harmonic functions on the plane with polynomial growth at most d to show that this family is essentially small.
Lipalpha harmonic approximation on closed sets
Proceedings of the American Mathematical Society, 2001
In this paper the Lipα harmonic approximation (0 < α < 1 2) on relatively closed subsets of a domain in the complex plane is characterized under the same conditions given by S. Gardiner for the uniform case. Thus, the result of P. Paramonov on Lipα harmonic polynomial approximation for compact subsets is extended to closed sets. Moreover, the problem of uniform approximation with continuous extension to the boundary for harmonic functions and similar questions in Lipα harmonic approximation are also studied.
Mathematische Nachrichten, 1995
We study the maximal function Mf(x) = sup I f (x + y, t)l when C2 is a region in the (Y , t)€O upper half space RY+' and f (x , t) is the harmonic extension to RY" of a distribution in the Besov space B;,,(RN) or in the Triebel-Lizorkin space FfJRN). In particular, we prove that when C2 = {lylN'(N-ap' < t < 1) the operator M is bounded from F;, ,(RN) into Lp(RN). The admissible regions for the spaces B;,.,(RN) with p < q are more complicated. a = N / p the region is { t > exp (-c J Y I~' (~-~)) } and has exponential contact. Finally, if p = 1 and a = N the region is (t > exp (-c lylPN)). These results have been extended to various settings. A. NAGEL and E. M. STEIN have considered, in particular, Bessel potentials of distributions in the Hardy spaces Hp(RN) with 0 < p < +a. P. AHERN and A. NAGEL have considered the boundary behavior of holomorphic and harmonic functions with respect to singular measures. J. R. DORRONSORO has studied the tangential convergence of Poisson integrals for certain spaces of regular functions which contain the Triebel-Lizorkin spaces F;JRN) with 1 < p < + 0 0 and 1 < q < + co. D. T. SHON has considered the boundary behavior of holomorphic functions 5 Math. Nachr.. Bd. 172
The 𝐿^{𝑝} Dirichlet problem and nondivergence harmonic measure
Transactions of the American Mathematical Society, 2002
We consider the Dirichlet problem \[ { L u a m p ; = 0 a m p ; a m p ; in D , u a m p ; = g a m p ; a m p ; on ∂ D \left \{ \begin {aligned} \mathcal {L} u & = 0 &\text {in DDD},\\ u &= g &\text {on partialD\partial DpartialD} \end {aligned} \right . \] for two second-order elliptic operators L k u = ∑ i , j = 1 n a k i , j ( x ) ∂ i j u ( x ) \mathcal {L}_k u=\sum _{i,j=1}^na_k^{i,j}(x)\,\partial _{ij} u(x) , k = 0 , 1 k=0,1 , in a bounded Lipschitz domain D ⊂ R n D\subset \mathbb {R}^n . The coefficients a k i , j a_k^{i,j} belong to the space of bounded mean oscillation BMO with a suitable small BMO modulus. We assume that L 0 {\mathcal {L}}_0 is regular in L p ( ∂ D , d σ ) L^p(\partial D, d\sigma ) for some p p , 1 > p > ∞ 1>p>\infty , that is, ‖ N u ‖ L p ≤ C ‖ g ‖ L p \|Nu\|_{L^p}\le C\,\|g\|_{L^p} for all continuous boundary data g g . Here σ \sigma is the surface measure on ∂ D \partial D and N u Nu is the nontangential maximal operator. The aim of this paper is to establis...