Multiplier (Fourier Analysis) - The Lp Boundedness Problem

The Lp Boundedness Problem

The boundedness problem (for any particular p) for a given group G is, stated simply, to identify the multipliers such that the corresponding multiplier operator is bounded from to . Such multipliers are usually simply referred to as " multipliers". Note that as multiplier operators are always linear, such operators are bounded if and only if they are continuous. This problem is considered to be extremely difficult in general, but many special cases can be treated. The problem depends greatly on p, although there is a duality relationship: if and, then a multiplier operator is bounded on if and only if it is bounded on .

The Riesz-Thorin theorem shows that if a multiplier operator is bounded on two different spaces, then it is also bounded on all intermediate spaces. Hence we get that the space of multipliers is smallest for and L∞ and grows as one approaches, which has the largest multiplier space.