a Holder space
/ HUL-der /
Continuity says a function does not jump; differentiability says it has a clean tangent line. Between these two there is a whole graded scale of in-between smoothness — functions that are continuous and even bend by a controlled amount, but not necessarily differentiable. Holder spaces measure exactly that intermediate regularity, putting a precise number on 'how smoothly' a function varies. They are the natural language for the classical (Schauder) side of elliptic theory, the counterpart to the integral-based Sobolev spaces.
A function u is Holder continuous with exponent alpha (a number between 0 and 1) if its variation is controlled by a fractional power of distance: |u(x) - u(y)| is at most C times |x - y|^alpha for all x, y, for some constant C. When alpha = 1 this is Lipschitz continuity (slope bounded); for alpha below 1 it allows steeper-than-linear local behaviour, like the square-root cusp |x|^{1/2}, which is Holder-1/2 but not Lipschitz. The space C^{k,alpha} then consists of functions that are k times continuously differentiable and whose k-th derivatives are Holder continuous with exponent alpha — a refined ruler that reads regularity 'between' the integer levels C^k and C^{k+1}. Its norm adds the sizes of all derivatives up to order k plus the Holder constant of the top derivative. Holder continuity is the pointwise, geometric cousin of Sobolev integrability; the Sobolev embedding theorem is precisely the dictionary that translates enough Sobolev derivatives into a Holder exponent.
Where they earn their keep: the Schauder estimates, the classical workhorse of elliptic regularity, are stated entirely in Holder spaces — if the data and coefficients of an elliptic equation are C^{k,alpha}, then the solution is C^{k+2,alpha}, gaining exactly two derivatives. This clean 'gain two' behaviour is cleaner in Holder spaces than in any L^p space, which is why the Schauder theory prefers them. The reason fractional exponents are essential, and integers alone will not do, is a genuine fact, not a convenience: the Newtonian potential of a merely bounded (L-infinity) source is generally NOT C^2 — its second derivatives can fail to be continuous — but it IS C^{1,alpha} and, if the source is Holder, C^{2,alpha}. Holder spaces are the precise setting where elliptic equations gain their two derivatives without exception, which integer-order C^k spaces cannot provide.
The function u(x) = |x|^{1/2} on (-1, 1) is continuous and Holder continuous with exponent 1/2: |u(x) - u(y)| stays within a constant times |x - y|^{1/2}. But it is NOT Lipschitz — its slope blows up like 1/(2 square root of |x|) near 0 — so it lives in C^{0,1/2} but not in C^{0,1}. The single number alpha = 1/2 records exactly how its cusp at the origin is shaped.
The exponent alpha pins down the shape of a cusp: |x|^{1/2} is Holder-1/2 but not Lipschitz.
The exponent alpha must be strictly between 0 and 1 in the genuine Holder range — alpha = 1 collapses to Lipschitz and 'alpha > 1' forces the function to be constant (the only functions with |u(x)-u(y)| at most C|x-y|^{1+} are constants), so there is no room above 1. And Holder regularity is NOT implied by mere continuity: continuous functions exist that are Holder of no positive exponent at all (badly oscillating ones), so 'continuous' is strictly weaker than 'Holder continuous'.