{'fraction_non_alphanumeric': 0.01569506726457399, 'fraction_numerical': 0.008968609865470852, 'mean_word_length': 5.478260869565218, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
The Multiplication Operator in Sobolev Spaces with Respect to Measures

We consider the multiplication operator, M, in Sobolev spaces with respect to general measures and give a characterization for M to be bounded, in terms of sequentially dominated measures. This has important consequences for the asymptotic behaviour of Sobolev orthogonal polynomials. Also, we study properties of Sobolev spaces with respect to measures. 2001 Academic Press