asking about quarter wavelength transformer?

can anyone tell me whether the microstrip quarter wavelength use guided wavelength or free-space wave length? i read some documents and some of them say it is guided and the others say it is free-space? please help me i'm so confused.

Guided wavelength. All this line impedance transformation and Smith chart stuff works on guided wavelength.
Free space parameters only matter when you have medium free space.

Thank you for your reply. SO, if i have an array antenna and the distance between elements is half wavelength, that "half wavelength" is guided wave length too, isn't it?

If you are referring to the array antenna characteristic, the answer is no. The characteristic is determined by the free space distance of elements.