The Weighted L p −L q Boundedness of Commutators of Schrödinger Type Operators on the Stratified Lie Group G

C-selfadjointness of the product of a composition operator and a maximal differentiation operator

“…The result in [18] is stronger than what we stated here and has been obtained in the Bloom setting but we are not concern with that here, and to the best of our knowledge these are all that are known about the operator norm of paraproducts. The ideas used in [18] are similar to those methods used in [17], where the author has completed the characterization of the operator norm of commutators of a non-degenerate Calderon-Zygmund operator, T , and pointwise multiplication with a locally integrable function b. In fact, it is shown in [17] that for 1 < p, q < ∞ we have:…”

“…A useful property of sparse families which is crucial for us here is that although they can have a lot of overlaps, they behave in a way as they are disjoint. A simple example of this phenomena is the following lemma which is quite well-known and its proof can be found in [17,18]. Lemma 2.2.…”

“…where in the above Lr (R n ) is usual L r (R n ) modulo constants and Ċα (R n ) is the homogeneous Holder space. The strategy of the proof, both in [17] and [18], is to bound the local mean oscillation of the symbol of the operator by testing it on some suitable test functions.…”

“…Here, although we are dealing with the operator from H p to H q , in all cases where p ≤ q our arguments are essentially the same as in [1], [17] and [18]. In order to show that the symbol belongs to BMO(R n ) or Λα (R n ), we bound an appropriate mean oscillation of the symbol of the operator.…”

Preparation of bi‐continuous poly(acrylonitrile‐co‐methyl acrylate) microporous membranes by a thermally induced phase separation method

“…When PAN is heated to near its melting point, the pyrolysis will occur and polycyclic aromatic side group forms as well as small molecules leaches out . Therefore, PAN porous membranes are commonly produced by a nonsolvent induced phase separation (NIPS) method at a relatively low temperature, and large amounts of polar, toxic, or corrosive solvents, for example, dimethyl formamide or dimethyl sulfoxide, which always leading to production costly and environment pollution, should be resolved. And after considering the factors of solvent type, polymer concentration, nonsolvent system type and composition, additives to the polymer solution, and film casting conditions, the defects will be conquered, which ultimately limit the development of PAN membranes.…”