Research Notes

I recently made a post about complex interpolation , which is a machine for taking two Banach spaces, $A$ and $B$, and spitting out a continuous family of "interpolating" Banach spaces $(A,B)_x.$ The properties of these spaces were: For any $v$ in the intersection $A \cap B$, one has a bound $$\lVert v \rVert_{(A,B)_x} \leq \lVert v \rVert_A^{1-x} \lVert v \rVert_B^x.$$ The above characterization descends to maps; i.e., for continuous maps $T : A_1 \to B_1$ and $T : A_2 \to B_2$ that are "compatible with the sum $A_1 + A_2$" in some technical sense, one obtains an interpolating bound $$\lVert T \rVert_{(A_1, A_2)_x \to (B_1, B_2)_x} \leq \lVert T \rVert_{A_1 \to B_1}^{1-x} \lVert T \rVert_{A_2 \to B_2}^{x}.$$ There's a completely different technique for creating "interpolating bounds" that seems to be well known to a certain class of mathematicians, but that I haven't seen used in physics. This is the method of "real interpolation." I...

"Interpolation" is a broad term for starting with two concrete bounds on a mathematical object, and producing a range of other bounds that continuously fill in "the abstract space of bounds" between the extreme points that you started with. A very typical example is provided by linear operators on the set of functions on a measure space. If $T$ is a linear map defined on simple functions, then there are all sorts of continuity properties $T$ might have, depending on which $L^p$ norm I pick for its domain and which $L^p$ norm I pick for its image. We can talk about the norm $$\lVert T \rVert_{p \to q} = \sup_{\lVert f \rVert_p = 1} \lVert T f \rVert_q,$$ and ask whether this is finite for a given choice of $p$ and $q,$ or more concretely compute this norm as a function of $(p,q).$ Interpolation essentially says: supposing you know $\lVert T \rVert_{p_1, q_1}$ and $\lVert T \rVert_{p_2, q_2}$, can you deduce an upper bound for a family of norms $\lVert T \rVert_{p_t, ...