Pointwise regularity criteria
For \(p\in (1,\infty)\), \(u>-d/p\) and \(x_0\in \mathbb{R}^d\) fixed, the Calderón-Zygmund class \(T^p_u(x_0)\) consists of those \(f\in L^p_{\text{ loc}}(\mathbb{R}^d)\) for which there is a polynomial \(P\) of degree less than \(u\) such that \[ \biggl(\frac{1}{r^d}\int_{B(x_0,r)} | f(x)-P(x-x_0)| ^p\, dx\bigg)^{1/p}\leq Cr^u \qquad(r>0). \] Suppose that one has a finite family \(\psi^{\epsilon}\), \(\epsilon \in E\), of compactly supported mother wavelets of suitable regularity that generate an orthonormal wavelet basis for \(L^2(\mathbb{R}^d)\). Let \(\psi_{I}^\epsilon\) be the \(L^\infty\) normalized dilate \(\psi^\epsilon_I(x) = \psi^\epsilon(2^jx-k)\) in which \(I=\prod_{i=1}^d 2^{-j}[k_i,\,k_i+1)\). One defines the local square function \[ S_f (j, x_0)=\left(\sum_{I\subset 3I_j(x_0); \,\varepsilon} | \langle f,\, \psi_I^\varepsilon\rangle| ^2 \chi_{I}(x)\right)^{1/2}. \] Here \(I_j(x_0)\) is the dyadic interval of sidelength \(2^{-j}\) containing \(x_0\), and the sum extends over all mother wavelet indices and over all dyadic subcubes of \(3I_j(x_0)\), the cube with the same center and three times the sidelength of \(I_j(x_0)\). It is proved that if \(f\in T^p_u(x_0)\), \(p\in (0,\infty)\), then there is a \(C\geq 0\) such that, for all \(j\geq 0\), \(\| S_f(j,x_0)\| _p\leq C 2^{-j(u+d/p)}\). When \(p=\infty\) the condition \(f\in T^\infty_u(x_0)\) becomes \[ \sum_{J\subset I;\, \epsilon} | J|\, | \langle f,\,\psi_J^\epsilon\rangle| ^2\leq C^2 2^{-uj}| I| \] whenever \(I\subset 3I(j,x_0)\) is dyadic. This condition is sharper than the usual two-microlocal condition.
scientific article; zbMATH DE number 2136414