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We consider local means with bounded smoothness for Besov-Morrey and Triebel-Lizorkin-Morrey spaces. Based on those we derive characterizations of these spaces in terms of Daubechies, Meyer, Bernstein (spline) and more general r-regular (father) wavelets, finally in terms of (biorthogonal) wavelets which can serve as molecules and local means, respectively. Hereby both, local means and wavelet decompositions satisfy natural conditions concerning smoothness and cancellation (moment conditions). Moreover, the given representations by wavelets are unique and yield isomorphisms between the considered function spaces and appropriate sequence spaces of wavelet coefficients. These wavelet representations lead to wavelet bases if, and only if, the function spaces coincide with certain classical Besov-Triebel-Lizorkin spaces.

We prove that for operators satistying weighted inequalities with A p weights the boundedness on a certain class of Morrey spaces holds with weights of the form |x| α w(x) for w ∈ A p . In the case of power weights the shift with respect to the range of Muckenhoupt weights was observed by N. Samko for the Hilbert transform, by H. Tanaka for the Hardy-Littlewood maximal operator, and by S. Nakamura and Y. Sawano for Calderón-Zygmund operators and others. We extend the class of weights and establish the results in a very general setting, with applications to many operators. For weak type Morrey spaces, we obtain new estimates even for the Hardy-Littlewood maximal operator. Moreover, we prove the necessity of certain A q condition.2010 Mathematics Subject Classification. Primary .

We prove that operators satisfying the hypotheses of the extrapolation theorem for Muckenhoupt weights are bounded on weighted Morrey spaces. As a consequence, we obtain at once a number of results that have been proved individually for many operators. On the other hand, our theorems provide a variety of new results even for the unweighted case because we do not use any representation formula or pointwise bound of the operator as was assumed by previous authors. To extend the operators to Morrey spaces we show different (continuous) embeddings of (weighted) Morrey spaces into appropriate Muckenhoupt A1 weighted Lp spaces, which enable us to define the operators on the considered Morrey spaces by restriction. In this way, we can avoid the delicate problem of the definition of the operator, often ignored by the authors. In dealing with the extension problem through the embeddings (instead of using duality) one is neither restricted in the parameter range of the p's (in particular p = 1 is admissible and applies to weak-type inequalities) nor the operator has to be linear. Another remarkable consequence of our results is that vector-valued inequalities in Morrey spaces are automatically deduced. On the other hand, we also obtain A∞-weighted inequalities with Morrey quasinorms.2010 Mathematics Subject Classification. Primary 42B35, 46E30, 42B15, 42B20; Secondary 42B25.

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